Fact-checked by Grok 2 weeks ago

Real analysis

Real analysis is the branch of mathematics that deals with limits, continuity, and related concepts for real-valued functions of a real variable, providing a rigorous foundation for calculus through the study of the properties of the real numbers and the behavior of sequences, series, and functions. It emphasizes precise definitions and proofs, such as the completeness axiom of the reals, which states that every nonempty subset of the real numbers that is bounded above has a least upper bound, ensuring the convergence of Cauchy sequences. Central to real analysis are the epsilon-delta definitions that formalize limits: a sequence (x_n) converges to x if for every \epsilon > 0, there exists N \in \mathbb{N} such that |x_n - x| < \epsilon for all n > N. This extends to function limits, where \lim_{x \to c} f(x) = L if for every \epsilon > 0, there exists \delta > 0 such that $0 < |x - c| < \delta implies |f(x) - L| < \epsilon. Continuity follows directly, with a function f continuous at c if \lim_{x \to c} f(x) = f(c), enabling theorems like the intermediate value theorem, which guarantees that continuous functions on closed intervals attain all values between their range endpoints. Differentiation in real analysis defines the derivative f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, leading to results such as Rolle's theorem—if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c \in (a, b) with f'(c) = 0—and the mean value theorem, which extends this to f'(c) = \frac{f(b) - f(a)}{b - a}. Integration focuses on the Riemann integral for bounded functions on closed intervals, where f is integrable if the upper and lower integrals coincide, connecting back to differentiation via the fundamental theorem of calculus, which asserts that if f is continuous and F(x) = \int_a^x f(t) \, dt, then F'(x) = f(x). These concepts underpin advanced topics like uniform convergence of function sequences and metric space generalizations, distinguishing real analysis as essential for pure and applied mathematics.

Foundations of the Real Numbers

Axiomatic construction of the reals

The axiomatic construction of the real numbers begins with the rational numbers \mathbb{Q}, which form an ordered field but lack completeness, allowing for gaps such as the irrationals. In 1872, Richard Dedekind provided one of the first rigorous constructions by defining real numbers as Dedekind cuts of the rationals. A Dedekind cut is a partition of \mathbb{Q} into two non-empty subsets A and B such that every element of A is less than every element of B, A has no greatest element, and A \cup B = \mathbb{Q}. The set of all such cuts forms the real numbers \mathbb{R}, where the lower set A represents the real number, and arithmetic operations are defined componentwise on the cuts (e.g., addition of cuts (A_1, B_1) and (A_2, B_2) yields (A_1 + A_2, B_1 + B_2)). Equivalence classes arise when two cuts define the same partition, ensuring uniqueness up to the order. Independently in the same year, Georg Cantor offered an alternative construction using Cauchy sequences of rationals. A Cauchy sequence \{q_n\}_{n=1}^\infty in \mathbb{Q} satisfies: for every \epsilon > 0 in \mathbb{Q}^+, there exists N \in \mathbb{N} such that |q_m - q_n| < \epsilon for all m, n > N. The real numbers \mathbb{R} are the equivalence classes of such sequences under the relation where \{q_n\} \sim \{r_n\} if \lim_{n \to \infty} (q_n - r_n) = 0 (i.e., the sequences are "null" apart). Addition and multiplication are defined pointwise on representatives, and the order is induced by comparing sequences eventually. This quotient construction embeds \mathbb{Q} densely into \mathbb{R}. Both constructions yield a system satisfying the axioms of the real numbers: the field axioms (commutative ring with unity under addition and multiplication, distributive, multiplicative inverses for non-zero elements), the order axioms (total order \leq compatible with addition and multiplication, i.e., trichotomy, transitivity, and positivity preserved), and the completeness axiom (every non-empty subset of \mathbb{R} that is bounded above has a least upper bound in \mathbb{R}, also known as the least upper bound property). These axioms characterize \mathbb{R} uniquely up to isomorphism, meaning any two systems satisfying them are order-isomorphic as fields. The resulting \mathbb{R} verifies the Archimedean property: for any x, y \in \mathbb{R} with x > 0, there exists n \in \mathbb{N} such that nx > y, which follows from the density of \mathbb{Q} in \mathbb{R} and the completeness axiom. Additionally, the rationals are dense in \mathbb{R}: between any two distinct reals a < b, there exists q \in \mathbb{Q} with a < q < b, inherited from the constructions where every real is a limit of rationals.

Algebraic and order properties

The real numbers \mathbb{R} form a field under the operations of addition and multiplication, satisfying the standard field axioms: addition and multiplication are commutative and associative, multiplication distributes over addition, there exists an additive identity 0 and multiplicative identity 1, every element has an additive inverse, and every nonzero element has a multiplicative inverse. Additionally, \mathbb{R} is equipped with a total order < that is compatible with these operations, making it an ordered field. The order axioms include: for all a, b \in \mathbb{R}, exactly one of a < b, a = b, or a > b holds (trichotomy); the order is transitive; if a < b, then a + c < b + c for any c \in \mathbb{R} (addition preserves order); and if a < b and c > 0, then ac < bc (multiplication by positives preserves order). The trichotomy property ensures that the order is total and excludes the possibility of incomparable elements, providing a linear structure to \mathbb{R}. The compatibility of the order with field operations implies monotonicity: adding a fixed real to both sides of an inequality preserves the inequality, and multiplying by a positive real does the same, while multiplying by a negative reverses it. These properties enable derivations of key inequalities from the ordered field structure. The absolute value function on \mathbb{R} is defined by |x| = x if x \geq 0 and |x| = -x if x < 0. It satisfies |x| \geq 0 for all x, |x| = 0 if and only if x = 0, |-x| = |x|, and |xy| = |x||y|. A fundamental inequality derived from the order axioms is the triangle inequality: for all x, y \in \mathbb{R}, |x + y| \leq |x| + |y|. This follows from considering cases based on the signs of x and y and applying the monotonicity of addition and multiplication. The rational numbers \mathbb{Q} are dense in \mathbb{R}: for any x, y \in \mathbb{R} with x < y, there exists r \in \mathbb{Q} such that x < r < y. To see this, if x \geq 0, the Archimedean property yields a natural number n > 1/(y - x), and then an integer m such that nx < m < ny, so r = m/n works; the case x < 0 reduces to the positive case by shifting. This density highlights the intimate algebraic interplay between rationals and reals within the ordered field. Basic inequalities like the arithmetic mean-geometric mean (AM-GM) inequality for two non-negative reals illustrate the power of the ordered structure. For x, y \geq 0, \frac{x + y}{2} \geq \sqrt{xy}, with equality if and only if x = y. The proof relies on monotonicity: \frac{x + y}{2} - \sqrt{xy} = \frac{(\sqrt{x} - \sqrt{y})^2}{2} \geq 0, since squares are non-negative. Such results underpin many applications in analysis by bounding sums and products via the order.

Completeness and topological structure

The completeness of the real numbers \mathbb{R} is captured by the least upper bound property, which states that every nonempty subset S \subseteq \mathbb{R} that is bounded above has a least upper bound \sup S \in \mathbb{R}. This axiom distinguishes \mathbb{R} from the rational numbers \mathbb{Q}, where subsets like \{q \in \mathbb{Q} \mid q^2 < 2\} lack a supremum within \mathbb{Q}. The property ensures that \mathbb{R} is complete as an ordered field, allowing for the existence of limits essential to analysis. This completeness axiom is equivalent to the monotone convergence theorem for sequences in ordered fields: every increasing sequence in \mathbb{R} that is bounded above converges to its supremum. To see the implication from the least upper bound property, consider an increasing bounded sequence \{x_n\}; let L = \sup\{x_n \mid n \in \mathbb{N}\}, then for any \epsilon > 0, there exists N such that x_N > L - \epsilon, and since the sequence is increasing, x_n \to L for n \geq N. The converse holds by constructing monotone sequences approximating the supremum of a bounded set. The standard topological structure on \mathbb{R} arises from the Euclidean metric d(x,y) = |x - y|, which generates the open sets as arbitrary unions of open intervals (a,b) = \{x \in \mathbb{R} \mid a < x < b\} with a < b. These open intervals form a basis for the topology, meaning every open set is a union of such intervals, and they satisfy the basis axioms: for any two basis elements (a,b) and (c,d) with x \in (a,b) \cap (c,d), there exists (e,f) contained in the intersection with x \in (e,f). This metric topology equips \mathbb{R} with a Hausdorff space where convergence of sequences aligns with the order structure via completeness. In this topology, the Heine-Borel theorem characterizes compactness: a subset K \subseteq \mathbb{R} is compact if and only if it is closed and bounded. Specifically, every closed bounded interval [a,b] is compact, as any open cover admits a finite subcover, though the full proof relies on the nested interval property derived from completeness (detailed later in the article). Complementarily, the Bolzano-Weierstrass theorem states that every bounded sequence in \mathbb{R} has a convergent subsequence, a direct consequence of completeness ensuring the existence of limit points in closed bounded sets. The real line \mathbb{R} admits homeomorphisms to itself via translations x \mapsto x + c for c \in \mathbb{R} and scalings x \mapsto kx for k > 0, which are affine transformations preserving the standard topology, open sets, and convergence. These maps are continuous bijections with continuous inverses, maintaining the metric up to scaling and thus the topological properties like compactness of bounded closed intervals.

Limits and Sequences

Limits of sequences

In real analysis, a sequence of real numbers \{x_n\}_{n=1}^\infty is said to converge to a limit L \in \mathbb{R} if for every \epsilon > 0, there exists a positive integer N such that for all n > N, |x_n - L| < \epsilon. This \epsilon-N definition captures the intuitive notion that the terms of the sequence eventually get arbitrarily close to L and stay there. The symbol \lim_{n \to \infty} x_n = L denotes this convergence. The limit of a convergent sequence, when it exists, is unique in the real numbers. Suppose \{x_n\} converges to both L and M; then for every \epsilon > 0, there exist N_1 and N_2 such that for n > \max(N_1, N_2), both |x_n - L| < \epsilon/2 and |x_n - M| < \epsilon/2, implying |L - M| \leq |L - x_n| + |x_n - M| < \epsilon. Since \epsilon > 0 is arbitrary, L = M. This uniqueness follows from the metric structure of \mathbb{R}, where the absolute value serves as the distance function. The algebra of limits provides rules for combining convergent sequences. If \{x_n\} \to L and \{y_n\} \to M, then \{x_n + y_n\} \to L + M and \{c x_n\} \to c L for any constant c \in \mathbb{R}. Additionally, \{x_n y_n\} \to L M. For quotients, if M \neq 0, then \{x_n / y_n\} \to L / M, provided y_n \neq 0 for sufficiently large n. These properties, proved using the \epsilon-N definition and triangle inequality, enable manipulation of limits much like algebraic operations on real numbers. A key result characterizing convergence is the monotone convergence theorem: every bounded monotone sequence of real numbers converges. Specifically, if \{x_n\} is increasing and bounded above, it converges to its least upper bound \sup \{x_n : n \in \mathbb{N}\}; if decreasing and bounded below, it converges to its greatest lower bound. This theorem relies on the completeness of \mathbb{R}, ensuring the supremum exists as a real number. Examples illustrate these concepts. The constant sequence x_n = c for all n, an arithmetic sequence with common difference zero, converges to c, as |x_n - c| = 0 < \epsilon holds for any N = 1. For geometric sequences, consider x_n = a r^{n-1} with |r| < 1; this converges to $0 because |x_n - 0| = |a| |r|^{n-1} \to 0 as n \to \infty, verifiable by choosing N > \frac{\log(\epsilon / |a|)}{\log |r|}. If |r| \geq 1, the sequence diverges unless a = 0.

Cauchy sequences and completeness

A sequence \{x_n\} in a metric space is called a Cauchy sequence if for every \epsilon > 0, there exists a positive integer N such that |x_m - x_n| < \epsilon for all integers m, n > N. This condition captures the idea that the terms of the sequence become arbitrarily close to each other as n increases, without initially specifying a particular limit point. Every Cauchy sequence is bounded, meaning there exists some M > 0 such that |x_n| \leq M for all n. In the real numbers \mathbb{R}, a sequence converges if and only if it is a Cauchy sequence. To see this, the forward direction follows from the definition of convergence, as a convergent sequence has terms approaching a fixed limit and thus getting close to each other. For the converse, if \{x_n\} is Cauchy in \mathbb{R}, it is bounded and thus has a convergent subsequence by the Bolzano-Weierstrass theorem; the full sequence then converges to the same limit using the Cauchy property to control distances. The completeness axiom of \mathbb{R}, often expressed via the least upper bound property, underpins this equivalence by ensuring that bounded Cauchy sequences converge. One constructive approach to finding the limit of a Cauchy sequence \{x_n\} in \mathbb{R} uses nested intervals: for each k \geq 1, define the interval I_k = [a_k, b_k] where a_k = \min\{x_n : n \geq k\} and b_k = \max\{x_n : n \geq k\}; these intervals are closed, bounded, and nested (I_{k+1} \subseteq I_k), with lengths tending to zero by the Cauchy condition, so their intersection is a single point, which is the limit. The rational numbers \mathbb{Q} lack this completeness property, as there exist Cauchy sequences in \mathbb{Q} that do not converge to any rational limit. For example, consider the sequence of rational approximations to \sqrt{2} obtained via Newton's method, such as x_1 = 1, x_{n+1} = \frac{1}{2}(x_n + \frac{2}{x_n}); this is Cauchy in \mathbb{Q} but converges in \mathbb{R} to the irrational \sqrt{2}, illustrating that \mathbb{Q} is incomplete. These properties establish \mathbb{R} as a complete metric space with the standard metric d(x, y) = |x - y|, meaning every Cauchy sequence in \mathbb{R} converges to a point in \mathbb{R}. This completeness is fundamental for subsequent developments in analysis, such as the existence of limits for continuous functions on closed intervals.

Limits of functions

In real analysis, the limit of a function f: D \to \mathbb{R} at a point a in the domain D, where a is a limit point of D, is defined using the epsilon-delta formalism to capture the behavior of f(x) as x approaches a without necessarily evaluating at a itself. Specifically, \lim_{x \to a} f(x) = L if and only if for every \epsilon > 0, there exists a \delta > 0 such that for all x \in D with $0 < |x - a| < \delta, it holds that |f(x) - L| < \epsilon. This definition ensures that f(x) can be made arbitrarily close to L by restricting x to a sufficiently small punctured neighborhood of a. An equivalent characterization of this limit uses sequences, bridging the concept to the convergence of sequences discussed earlier. The sequential criterion states that \lim_{x \to a} f(x) = L if and only if, for every sequence (x_n) in D with x_n \neq a and \lim_{n \to \infty} x_n = a, it follows that \lim_{n \to \infty} f(x_n) = L. This equivalence allows proofs involving limits of functions to leverage sequential arguments, providing a powerful tool for verification. Limits can also be defined from one side when approaching a. The right-hand limit \lim_{x \to a^+} f(x) = L exists if for every \epsilon > 0, there is a \delta > 0 such that a < x < a + \delta implies |f(x) - L| < \epsilon, assuming such x are in D; similarly for the left-hand limit \lim_{x \to a^-} f(x) = L with a - \delta < x < a. The two-sided limit exists only if both one-sided limits exist and are equal. For behavior as x grows without bound, the limit \lim_{x \to \infty} f(x) = L means that for every \epsilon > 0, there exists M > 0 such that if x > M, then |f(x) - L| < \epsilon; an analogous definition holds for \lim_{x \to -\infty} f(x) = L. Infinite limits describe unbounded growth: \lim_{x \to a} f(x) = \infty if for every M > 0, there exists \delta > 0 such that $0 < |x - a| < \delta implies f(x) > M; similar definitions apply for \lim_{x \to a} f(x) = -\infty, or for limits at \pm \infty. Basic algebraic operations preserve limits under suitable conditions. If \lim_{x \to a} f(x) = L and \lim_{x \to a} g(x) = M, then \lim_{x \to a} [f(x) + g(x)] = L + M, \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M, and if M \neq 0, \lim_{x \to a} [f(x)/g(x)] = L/M; these extend to scalar multiples and hold similarly for one-sided or infinite limits.

Continuity

Definition and basic properties of continuous functions

In real analysis, a function f: D \to \mathbb{R}, where D \subseteq \mathbb{R}, is said to be continuous at a point a \in D if \lim_{x \to a} f(x) = f(a). A function is continuous on a set if it is continuous at every point in that set; for instance, continuity on an interval means the function is continuous at each point within the interval. An equivalent characterization of continuity uses sequences: f is continuous at a if and only if, for every sequence (x_n) in D with x_n \to a, it follows that f(x_n) \to f(a). This sequential criterion highlights continuity as a property preserved under limits of sequences approaching the point. One fundamental property is the stability under composition: if g is continuous at a and f is continuous at g(a), then the composition f \circ g is continuous at a. This follows from the limit definition, as \lim_{x \to a} (f \circ g)(x) = f\left( \lim_{x \to a} g(x) \right) = f(g(a)). Classic examples of continuous functions include polynomials, which are continuous at every point in \mathbb{R} due to their finite sums and products of continuous identity functions. Rational functions, being quotients of polynomials, are continuous wherever the denominator is nonzero. Continuous functions also preserve order in the sense that if f is monotonic (say, nondecreasing) on an interval I, then f(I) is also an interval. For a strictly increasing continuous f on [a, b], f maps [a, b] onto [f(a), f(b)], maintaining the order of points. A key theorem illustrating these properties is the Intermediate Value Theorem: if f is continuous on the closed interval [a, b] and k lies between f(a) and f(b), then there exists c \in [a, b] such that f(c) = k. To prove this, assume without loss of generality that f(a) < k < f(b). Construct nested closed intervals [x_n, y_n] \subseteq [a, b] via bisection: start with [x_0, y_0] = [a, b], and at each step, let m_n be the midpoint; if f(m_n) < k, set [x_{n+1}, y_{n+1}] = [m_n, y_n], else [x_{n+1}, y_{n+1}] = [x_n, m_n]. This ensures f(x_n) < k < f(y_n) and x_n \leq y_n. By the nested interval property, which stems from the completeness of \mathbb{R} (the least upper bound property), there exists c \in \bigcap_n [x_n, y_n]. Continuity of f at c implies f(c) = k, as f(x_n) \to f(c) and f(y_n) \to f(c), squeezing k between them.

Uniform continuity

A function f: S \to \mathbb{R}, where S \subset \mathbb{R}, is said to be uniformly continuous on S if for every \epsilon > 0, there exists a \delta > 0 (independent of the location in S) such that for all x, y \in S with |x - y| < \delta, it holds that |f(x) - f(y)| < \epsilon. This condition strengthens the pointwise notion of continuity by requiring the \delta to work uniformly across the entire domain S, rather than depending on specific points. A fundamental result in real analysis establishes that continuity on a compact set implies uniform continuity. Specifically, if K \subset \mathbb{R} is compact and f: K \to \mathbb{R} is continuous, then f is uniformly continuous on K. In \mathbb{R}, compactness of K is equivalent to K being closed and bounded by the Heine-Borel theorem. To prove the result, assume for contradiction that f is not uniformly continuous. Then there exists \epsilon_0 > 0 such that for every n \in \mathbb{N}, there are points x_n, y_n \in K with |x_n - y_n| < 1/n but |f(x_n) - f(y_n)| \geq \epsilon_0. The sequences (x_n) and (y_n) are in the compact set K, so by sequential compactness, they each have convergent subsequences converging to the same limit z \in K (since |x_n - y_n| \to 0). Continuity of f at z then implies |f(x_{n_k}) - f(y_{n_k})| \to 0 along the subsequence, contradicting the choice of \epsilon_0. Thus, f must be uniformly continuous. Not all continuous functions on non-compact sets are uniformly continuous, as illustrated by the function f(x) = 1/x on the open interval (0, 1). This function is continuous on (0, 1) because the reciprocal is well-defined and the limit exists at each point in the domain. However, it fails to be uniformly continuous: consider sequences x_n = 1/n and y_n = 1/(n+1) for n \in \mathbb{N}. Then |x_n - y_n| = |1/n - 1/(n+1)| = 1/(n(n+1)) \to 0, but |f(x_n) - f(y_n)| = |n - (n+1)| = 1 \not\to 0. For \epsilon = 1/2, no single \delta > 0 works for all pairs near 0, as the function's slope becomes arbitrarily steep. Uniform continuity has useful extensions, particularly regarding sequences. If f is uniformly continuous on an interval I \subset \mathbb{R}, then it maps Cauchy sequences in I to Cauchy sequences in \mathbb{R}. To see this, let \{x_n\} be Cauchy in I, so for every \epsilon > 0, there exists N \in \mathbb{N} such that |x_m - x_n| < \delta for m, n > N, where \delta > 0 is chosen from the uniform continuity of f for this \epsilon. Then |f(x_m) - f(x_n)| < \epsilon for m, n > N, making \{f(x_n)\} Cauchy. A stronger condition than uniform continuity is Lipschitz continuity: a function f: S \to \mathbb{R} is Lipschitz continuous on S if there exists a constant K \geq 0 such that |f(x) - f(y)| \leq K |x - y| for all x, y \in S. This implies uniform continuity, since for any \epsilon > 0, choosing \delta = \epsilon / K (if K > 0) ensures |f(x) - f(y)| < \epsilon whenever |x - y| < \delta. If K = 0, then f is constant and trivially uniform.

Absolute continuity

A function f: [a, b] \to \mathbb{R} is said to be absolutely continuous if for every \varepsilon > 0, there exists a \delta > 0 such that for any finite collection of disjoint subintervals (a_i, b_i) of [a, b] satisfying \sum |b_i - a_i| < \delta, it holds that \sum |f(b_i) - f(a_i)| < \varepsilon. This condition strengthens uniform continuity by controlling the total oscillation of f not just by the total length of intervals, but in a way that accounts for the function's behavior across disjoint parts, making it particularly suited for integration theory. Absolutely continuous functions are intimately connected to integration: a function f on [a, b] is absolutely continuous if and only if it can be expressed as f(x) = f(a) + \int_a^x g(t) \, dt for some integrable function g, where the integral may be taken in the Riemann or Lebesgue sense. This representation implies that f is the indefinite integral of its derivative, which exists almost everywhere, highlighting the role of absolute continuity in linking differentiation and integration on the real line. Absolute continuity implies that the function has bounded variation. The total variation of f on [a, b] is defined as V_f(a, b) = \sup \sum |f(x_{i+1}) - f(x_i)|, where the supremum is taken over all partitions a = x_0 < x_1 < \cdots < x_n = b of [a, b]. Functions of bounded variation can be decomposed into absolutely continuous and singular parts, but absolute continuity ensures the total variation is finite and controlled by the integral of the derivative's absolute value. In contrast, uniform continuity is a weaker property that does not necessarily imply bounded variation or this integral representation. Singular functions provide counterexamples to the converse: the Cantor function, also known as the Devil's staircase, is continuous and monotonically increasing on [0, 1], hence of bounded variation, but it is not absolutely continuous because its derivative is zero almost everywhere while the function increases from 0 to 1. This function maps the Cantor set, which has Lebesgue measure zero, onto an interval of positive measure, illustrating a singular component that violates absolute continuity. Absolutely continuous functions satisfy Lusin's condition (N), meaning they map sets of Lebesgue measure zero to sets of Lebesgue measure zero. This property underscores their preservation of null sets under the induced measure, distinguishing them from singular functions like the Cantor function, which fail condition (N).

Compactness and Connectedness

Compact sets

In real analysis, a subset K \subseteq \mathbb{R} is defined as compact if every open cover of K admits a finite subcover. An open cover consists of a collection of open sets \{U_\alpha : \alpha \in A\} such that K \subseteq \bigcup_{\alpha \in A} U_\alpha, and a finite subcover is a finite subfamily whose union still contains K. This topological notion captures the idea of "finiteness" in infinite settings, generalizing properties of finite sets. In the metric space \mathbb{R}, compactness is equivalent to sequential compactness: every sequence in K has a convergent subsequence with limit in K. This equivalence holds more generally for metric spaces, where the open cover definition implies sequential compactness via limit points of sequences, and sequential compactness implies compactness using countable bases and nested closed sets. For instance, if K is sequentially compact, any open cover can be refined to yield a finite subcover by extracting convergent subsequences and covering limit points. The Heine-Borel theorem characterizes compactness in \mathbb{R}: a subset K \subseteq \mathbb{R} is compact if and only if it is closed and bounded. This result, named after Eduard Heine and Émile Borel with foundational work by Bernard Bolzano, is fundamental to real analysis. To prove the forward direction (compact implies closed and bounded), compactness yields closedness since complements of closed sets are open unions, and boundedness follows from the cover \{(-n, n) : n \in \mathbb{N}\}, which has a finite subcover containing K. For the converse (closed and bounded implies compact), assume K is unbounded; then the cover \{(-n, n) : n \in \mathbb{N}\} has no finite subcover, a contradiction. If bounded but not closed, sequences converging outside K lack subsequences in K. The full proof for bounded closed sets uses the nested interval theorem: suppose an open cover \{U_\alpha\} has no finite subcover of [a, b] \supseteq K; start with I_1 = [a, b] and at each step bisect into two halves, selecting as I_{n+1} the half that admits no finite subcover from \{U_\alpha\} (at least one such half exists, as otherwise the whole would have one); the nested closed intervals I_n have intersection point x \in K, which lies in some U_{\alpha_0}; openness of U_{\alpha_0} ensures some I_n \subseteq U_{\alpha_0}, contradicting that I_n has no finite subcover. Compact sets in \mathbb{R} are necessarily closed and bounded, as per Heine-Borel. Finite unions of compact sets are compact: if K_1, \dots, K_m are compact, any open cover of \bigcup K_i restricts to finite subcovers for each K_i, combining finitely. Continuous images of compact sets are compact, implying boundedness and closedness under continuous maps. A key consequence is the extreme value theorem: if f: K \to \mathbb{R} is continuous and K \subseteq \mathbb{R} is compact, then f attains its maximum and minimum values on K. This follows since f(K) is compact (hence closed and bounded), so \sup f(K) and \inf f(K) are achieved. For example, on [a, b], continuous functions reach extrema, enabling bounds in analysis. Examples of compact sets include closed bounded intervals [a, b], which satisfy Heine-Borel directly. Finite sets are compact, as any cover has a finite subcover by selecting sets containing each point. In contrast, the open interval (0, 1) is not compact: the cover \{(1/n, 1) : n = 2, 3, \dots \} has no finite subcover, as any finite collection misses points near 0. Similarly, \mathbb{R} is not compact, covered without finite subcover by \{(-n, n) : n \in \mathbb{N}\}. The set \{1/n : n \in \mathbb{N}\} \cup \{0\} is compact, while without 0 it is not, as the sequence $1/n has no convergent subsequence in the set.

Connected sets and intervals

In the context of real analysis, a subset C \subseteq \mathbb{R} is defined to be connected if it is not the union of two nonempty disjoint relatively open sets in the subspace topology induced from \mathbb{R}. This means that there do not exist nonempty subsets A, B \subseteq C such that A \cup B = C, A \cap B = \emptyset, and both A and B are open in the relative topology on C. A fundamental characterization in \mathbb{R} states that a subset is connected if and only if it is an interval, which may be open, closed, half-open, a ray (bounded on one side), a singleton, or the entire line \mathbb{R}. To prove this, first note that any interval is connected: suppose an interval I is disconnected, written as I = U \cup V with U, V nonempty, disjoint, and relatively open; assume without loss of generality that b \in V and let c = \sup U; since U is closed in the subspace (as complement of open V), c \in U; but V relatively open implies an interval (b - \epsilon, b] \cap I \subseteq V, so c < b, and U relatively open implies [c, c + \delta) \cap I \subseteq U for some \delta > 0, contradicting c = \sup U. Conversely, if C \subseteq \mathbb{R} is connected and nonempty with at least two points, let c \in C and consider the sets A = \{x \in C : x < c\} and B = \{x \in C : x > c\}; if both are nonempty, let s = \sup A; by contradiction, s \notin C would separate C into relatively open sets around points less than and greater than s, violating connectedness, so s \in C and C must fill the interval between its infimum and supremum. As a direct consequence, the continuous image of a connected set is connected, yielding the intermediate value theorem: if f: [a, b] \to \mathbb{R} is continuous and k lies between f(a) and f(b), then there exists c \in [a, b] such that f(c) = k, since f([a, b]) is a connected interval containing f(a) and f(b). In \mathbb{R}, path-connectedness—where any two points can be joined by a continuous path, such as the straight-line segment—coincides with connectedness for subsets, as every connected subset is an interval and thus path-connected via linear parametrization. Examples of disconnected sets include the rational numbers \mathbb{Q}, which are totally disconnected, meaning their only connected subsets are singletons; for any two distinct p, q \in \mathbb{Q} with p < q, there exists an irrational r \in (p, q), separating \mathbb{Q} into relatively open sets \mathbb{Q} \cap (-\infty, r) and \mathbb{Q} \cap (r, \infty). Another is the Cantor set, a compact totally disconnected perfect set in [0, 1]; constructed by iteratively removing middle thirds, it has no intervals and can be separated at any two points by the construction's open intervals.

Differentiation

Definition of the derivative

The derivative of a function f: I \to \mathbb{R}, where I is an open interval containing a \in \mathbb{R}, at the point a is defined as f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}, provided the limit exists as a real number. This limit quantifies the instantaneous rate of change of f at a. Geometrically, if the graph of f is considered, f'(a) represents the slope of the tangent line to the graph at the point (a, f(a)). A function f is differentiable at a if f'(a) exists. Differentiability at a implies continuity of f at a. To see this, note that f(a + h) - f(a) = h \cdot \frac{f(a + h) - f(a)}{h}. As h \to 0, the right-hand side tends to $0 because \frac{f(a + h) - f(a)}{h} \to f'(a) (a finite number) and h \to 0, so f(a + h) \to f(a). The derivative satisfies basic algebraic properties when f and g are differentiable at a and c \in \mathbb{R}. The sum rule states that (f + g)'(a) = f'(a) + g'(a). The scalar multiple rule gives (c f)'(a) = c f'(a). The product rule is (f g)'(a) = f'(a) g(a) + f(a) g'(a). For the chain rule, if g is differentiable at a and f is differentiable at g(a), then (f \circ g)'(a) = f'(g(a)) g'(a). Rolle's theorem, a special case of the mean value theorem, states that if f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) with f(a) = f(b), then there exists c \in (a, b) such that f'(c) = 0. The mean value theorem generalizes this: if f is continuous on [a, b] and differentiable on (a, b), then there exists c \in (a, b) such that f'(c) = \frac{f(b) - f(a)}{b - a}. Geometrically, this asserts that the secant line slope over [a, b] equals the tangent line slope at some interior point. Examples illustrate these concepts. For a polynomial f(x) = \sum_{k=0}^n a_k x^k, the derivative is f'(x) = \sum_{k=1}^n k a_k x^{k-1}, obtained by applying the algebraic rules term by term. For the exponential function defined by e^x = \lim_{n \to \infty} (1 + x/n)^n, the derivative is f'(x) = e^x, verifiable using the definition and properties of limits.

Mean value theorem and applications

The mean value theorem (MVT) states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c \in (a, b) such that f'(c) = \frac{f(b) - f(a)}{b - a}. This theorem interprets the derivative as the slope of the secant line connecting the endpoints of the interval. To prove the MVT, first consider Rolle's theorem, which asserts that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c \in (a, b) with f'(c) = 0. The proof of Rolle's theorem relies on the extreme value theorem: since f attains its maximum or minimum on the compact set [a, b], if the extremum is interior, the derivative vanishes there by the definition of differentiability; otherwise, if at an endpoint, f(a) = f(b) forces an interior point where the derivative is zero. For the MVT, define an auxiliary function g(x) = f(x) - f(a) - \frac{f(b) - f(a)}{b - a}(x - a). Then g(a) = g(b) = 0, so by Rolle's theorem applied to g, there exists c \in (a, b) with g'(c) = 0, which simplifies to f'(c) = \frac{f(b) - f(a)}{b - a}. This proof assumes the extreme value theorem on compact sets, ensuring the existence of extrema. A key application of the MVT is to monotonicity. If f'(x) \geq 0 for all x \in (a, b), then for any x_1 < x_2 in [a, b], the MVT implies f(x_2) - f(x_1) = f'(c)(x_2 - x_1) \geq 0 for some c \in (x_1, x_2), so f is increasing on [a, b]. If instead f'(x) > 0 on (a, b), then f is strictly increasing, as the difference f(x_2) - f(x_1) > 0. The MVT also underlies L'Hôpital's rule for evaluating limits. Suppose \lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0 or \infty, with f and g differentiable on an interval around a (except possibly at a), g'(x) \neq 0, and \lim_{x \to a} \frac{f'(x)}{g'(x)} = L. Then \lim_{x \to a} \frac{f(x)}{g(x)} = L, provided the latter limit exists. The proof uses Cauchy's mean value theorem, a generalization of the MVT: for continuous f, g on [a, b] and differentiable on (a, b) with g' \neq 0, there exists c \in (a, b) such that \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}. Applying this iteratively or in limit form yields the result. For convexity, consider twice differentiable functions. If f''(x) \geq 0 on an interval I, then f'(x) is increasing on I by the MVT applied to f', implying f is convex: for x, y \in I and \lambda \in [0, 1], f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1 - \lambda) f(y). Equivalently, the graph lies above its tangent lines, as the MVT shows the secant slope exceeds the left tangent slope and is below the right one. For twice differentiable functions, Jensen's inequality follows similarly, reinforcing convexity. Bernoulli's inequality, stating that (1 + x)^r \geq 1 + r x for r \geq 1 and x \geq -1, is a special case proved via the MVT. Consider f(t) = (1 + t)^r on [0, x]; then f(x) - f(0) = f'(c) x for some c \in (0, x), so (1 + x)^r - 1 = r (1 + c)^{r-1} x \geq r x since (1 + c)^{r-1} \geq 1. While the MVT requires only differentiability, not continuous differentiability, counterexamples exist where a function is differentiable everywhere but its derivative is discontinuous. For instance, define f(x) = x^2 \sin(1/x) for x \neq 0 and f(0) = 0; then f'(0) = 0 and f'(x) = 2x \sin(1/x) - \cos(1/x) for x \neq 0, but \lim_{x \to 0} f'(x) does not exist due to the oscillating \cos(1/x) term. Thus, f is differentiable on \mathbb{R} but f' is not continuous at 0.

Higher-order derivatives and Taylor's theorem

Higher-order derivatives of a function f are obtained by iteratively applying the differentiation operator. The first derivative is denoted f'(x) or \frac{df}{dx}, the second derivative f''(x) or \frac{d^2f}{dx^2}, and in general, the nth derivative f^{(n)}(x) for n \geq 1. If f possesses derivatives of all orders on an interval (a, b), it is said to be infinitely differentiable, or C^\infty(a, b). Taylor's theorem provides a polynomial approximation for a function near a point a, generalizing the mean value theorem to higher orders. Specifically, if f is (n+1)-times differentiable on an interval containing a and x, then f(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k + R_n(x), where the remainder R_n(x) satisfies the Lagrange form R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1} for some c between a and x. This theorem can be established by repeated application of the mean value theorem. An alternative expression for the remainder is the Peano form, which states that R_n(x) = o(|x - a|^n) as x \to a. This little-o notation emphasizes the local approximation error vanishing faster than the nth power of the distance from a. A classic example is the Taylor expansion of \sin x around a = 0: \sin x = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k+1}, where the finite partial sums approximate \sin x and the remainder tends to zero for all x as n \to \infty. For the infinite Taylor series to equal the function exactly on an interval, the remainder must satisfy R_n(x) \to 0 as n \to \infty for each x in that interval; functions satisfying this condition are called analytic. The theorem is named after Brook Taylor, who introduced finite expansions of this form in his 1715 work Methodus incrementorum directa et inversa.

Infinite Series

Convergence tests for series

A series \sum_{n=1}^\infty a_n of real numbers is said to converge if the sequence of its partial sums s_n = \sum_{k=1}^n a_k converges to a finite limit as n \to \infty. This definition, formalized by Augustin-Louis Cauchy, extends the notion of sequence convergence to infinite summations. The series converges absolutely if \sum_{n=1}^\infty |a_n| converges, which implies ordinary convergence but not conversely; conditionally convergent series converge without their absolute counterparts doing so. For series with nonnegative terms, the comparison test provides a fundamental criterion: if $0 \leq a_n \leq b_n for all sufficiently large n and \sum b_n converges, then \sum a_n converges; conversely, if \sum a_n diverges, then \sum b_n diverges. This test, rooted in Cauchy's foundational work on limits, relies on the monotonicity of partial sums for nonnegative sequences. A variant, the limit comparison test, applies when direct bounds are unavailable: for positive terms a_n and b_n, if \lim_{n \to \infty} a_n / b_n = L where $0 < L < \infty, then \sum a_n and \sum b_n either both converge or both diverge. This extension facilitates comparisons with known series like the p-series \sum 1/n^p, which converges for p > 1 and diverges for p \leq 1./09:_Sequences_and_Series/9.03:_The_Divergence_Test_and_p-Series) The ratio test assesses absolute convergence by examining \lim_{n \to \infty} |a_{n+1}/a_n| = L: if L < 1, the series converges absolutely; if L > 1, it diverges; if L = 1, the test is inconclusive./09:_Sequences_and_Series/9.06:_Ratio_and_Root_Tests) First published by Jean le Rond d'Alembert in 1768 and later refined by Cauchy, this test compares the series to a geometric one, succeeding when terms grow or decay exponentially. Similarly, the root test uses \limsup_{n \to \infty} |a_n|^{1/n} = L: absolute convergence holds for L < 1, divergence for L > 1, and inconclusiveness for L = 1./09:_Sequences_and_Series/9.06:_Ratio_and_Root_Tests) Introduced by Cauchy in his 1821 Cours d'analyse, it is particularly effective for series where nth roots reveal asymptotic behavior more clearly than ratios. For alternating series \sum_{n=1}^\infty (-1)^{n+1} b_n with b_n > 0, the Leibniz test (or alternating series test) states that the series converges if b_n is monotonically decreasing and \lim_{n \to \infty} b_n = 0. This criterion, due to Gottfried Wilhelm Leibniz in the late 17th century, guarantees conditional convergence when absolute convergence fails, as the partial sums oscillate but approach a limit bounded by the first omitted term. A classic example of divergence is the harmonic series \sum_{n=1}^\infty 1/n, which can be shown to diverge using the integral test: since the function f(x) = 1/x is positive, continuous, and decreasing for x \geq 1, and \int_1^\infty dx/x = \infty, the series diverges. This test, formalized by Cauchy, links discrete sums to continuous integrals. In contrast, the geometric series \sum_{n=0}^\infty r^n converges absolutely to $1/(1-r) for |r| < 1, a result known since antiquity but rigorously established in the context of infinite series by Euler and others in the 18th century.

Power series and radius of convergence

A power series centered at a point a \in \mathbb{R} is an infinite series of the form \sum_{n=0}^{\infty} c_n (x - a)^n, where c_n are real coefficients. Every such series has a radius of convergence R, where $0 \leq R \leq \infty, such that the series converges absolutely for all x satisfying |x - a| < R and diverges for |x - a| > R. The radius R can be determined using the root test formula \frac{1}{R} = \limsup_{n \to \infty} |c_n|^{1/n}, or, when the limit exists, the ratio test formula R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|. The interval of convergence is the open interval (a - R, a + R), but convergence at the endpoints x = a \pm R must be checked separately using standard series tests, as the behavior there is not guaranteed by the radius alone. Within the interval of convergence, the power series converges uniformly on any compact subinterval [a - r, a + r] where $0 \leq r < R. If a power series converges on the open interval (a - R, a + R), then the series obtained by termwise differentiation also converges on the same interval and has the same radius R, with the derivative of the sum equal to the sum of the derivatives: \frac{d}{dx} \sum_{n=0}^{\infty} c_n (x - a)^n = \sum_{n=1}^{\infty} n c_n (x - a)^{n-1}. Similarly, termwise integration preserves the radius of convergence, and the integral of the sum equals the sum of the integrals: \int \sum_{n=0}^{\infty} c_n (x - a)^n \, dx = \sum_{n=0}^{\infty} \frac{c_n}{n+1} (x - a)^{n+1} + C. Functions represented by power series within their interval of convergence are analytic, meaning they are infinitely differentiable, and the coefficients satisfy c_n = \frac{f^{(n)}(a)}{n!} for the sum function f. This connection implies that such functions are twice continuously differentiable (in fact, C^\infty) inside the radius. For example, the exponential series \sum_{n=0}^{\infty} \frac{x^n}{n!} has radius R = \infty by the ratio test, converging for all real x to e^x. Another example is the series for the natural logarithm, \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}, which has radius R = 1 via the root test and converges to \log(1 + x) for |x| < 1, with conditional convergence at x = 1.

Fourier series

Fourier series provide a method to represent periodic functions as infinite sums of sines and cosines, leveraging the orthogonality of the trigonometric system on the interval [- \pi, \pi]. For a $2\pi-periodic function f: \mathbb{R} \to \mathbb{R}, the Fourier series is given by f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos(nx) + b_n \sin(nx) \right), where the coefficients are a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) \, dx \quad (n \geq 0), \quad b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) \, dx \quad (n \geq 1). These formulas arise from the orthogonality relations of the trigonometric functions over [- \pi, \pi], specifically \int_{-\pi}^\pi \cos(mx) \cos(nx) \, dx = \begin{cases} \pi \delta_{mn} & m, n \geq 1, \\ 2\pi & m = n = 0, \\ 0 & m \neq n, \end{cases} \int_{-\pi}^\pi \sin(mx) \sin(nx) \, dx = \pi \delta_{mn} \quad (m, n \geq 1), \quad \int_{-\pi}^\pi \sin(mx) \cos(nx) \, dx = 0, where \delta_{mn} is the Kronecker delta. By projecting f onto these basis functions and using the integrals to compute inner products, the coefficients isolate each harmonic component. The pointwise convergence of the Fourier series is governed by Dirichlet's theorem, which states that if f is piecewise continuous on [-\pi, \pi] with a finite number of finite discontinuities and finite variation (i.e., piecewise smooth), then the series converges at each x to f(x) where f is continuous, and to the average \frac{f(x^+) + f(x^-)}{2} at points of jump discontinuity. This result, originally established by Peter Gustav Lejeune Dirichlet in 1829, ensures representation for a broad class of practical functions, such as those arising in physical applications. Near discontinuities, however, the partial sums exhibit overshoot known as the Gibbs phenomenon, where the approximation oscillates and exceeds the target value by approximately 8.9% of the jump height, regardless of the number of terms included. This ringing effect, first noted by Henry Wilbraham in 1848 and later analyzed by Josiah Willard Gibbs, stems from the slow decay of high-frequency coefficients and nonuniform convergence at jumps. In the mean-square sense, the Fourier series converges to f for square-integrable functions on [-\pi, \pi], as captured by Parseval's identity: \frac{1}{\pi} \int_{-\pi}^\pi |f(x)|^2 \, dx = \frac{a_0^2}{2} + \sum_{n=1}^\infty (a_n^2 + b_n^2). This equality equates the L^2-norm of f to the \ell^2-norm of its coefficients, reflecting the completeness of the trigonometric basis in L^2([-\pi, \pi]). A classic example is the square wave function f(x) = \operatorname{sgn}(\sin x), which jumps between -1 and $1with period2\pi$. Being odd, its Fourier series contains only sine terms: f(x) = \frac{4}{\pi} \sum_{k=1,3,5,\ldots}^\infty \frac{1}{k} \sin(kx), with coefficients b_k = 4/(k\pi) for odd k and zero otherwise. This series sums odd harmonics, illustrating how discontinuities lead to Gibbs overshoot near x = 0, \pi.

Integration

Riemann integral

The Riemann integral provides a method to define the definite integral of a bounded real-valued function f on a closed and bounded interval [a, b]. Consider a partition P = \{x_0 = a, x_1, \dots, x_n = b\} of [a, b], where each subinterval has length \Delta x_i = x_i - x_{i-1}. For each subinterval [x_{i-1}, x_i], let M_i = \sup \{f(x) : x \in [x_{i-1}, x_i]\} be the supremum of f and m_i = \inf \{f(x) : x \in [x_{i-1}, x_i]\} be the infimum. The upper Darboux sum is U(P, f) = \sum_{i=1}^n M_i \Delta x_i, and the lower Darboux sum is L(P, f) = \sum_{i=1}^n m_i \Delta x_i. The upper integral is \overline{\int_a^b} f(x) \, dx = \inf \{U(P, f) : P \text{ partition of } [a, b]\}, and the lower integral is \underline{\int_a^b} f(x) \, dx = \sup \{L(P, f) : P \text{ partition of } [a, b]\}. A bounded function f on [a, b] is Riemann integrable if and only if the upper and lower integrals are equal, in which case the Riemann integral is defined as \int_a^b f(x) \, dx = \overline{\int_a^b} f(x) \, dx = \underline{\int_a^b} f(x) \, dx. Every continuous function on [a, b] is Riemann integrable. Additionally, a bounded function on [a, b] with only finitely many discontinuities is Riemann integrable. The Riemann integral satisfies several fundamental properties. It is linear: if f and g are Riemann integrable on [a, b] and \alpha, \beta \in \mathbb{R}, then \alpha f + \beta g is Riemann integrable and \int_a^b (\alpha f(x) + \beta g(x)) \, dx = \alpha \int_a^b f(x) \, dx + \beta \int_a^b g(x) \, dx. It also respects the order of integration limits: \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx. A key connection to differentiation is given by the first part of the fundamental theorem of calculus: if f is continuous on [a, b], then the function F(x) = \int_a^x f(t) \, dt is differentiable on (a, b) with F'(x) = f(x), and F is continuous on [a, b]. For example, the function f(x) = x^2 on [a, b] is continuous and thus Riemann integrable, with \int_a^b x^2 \, dx = \frac{b^3 - a^3}{3}. Bounded step functions, which are constant on finitely many subintervals of [a, b], are also Riemann integrable; for such a function, the integral equals the sum of the products of the constant values and the lengths of the corresponding subintervals.

Fundamental theorems of calculus

The fundamental theorems of calculus comprise two key results that link the operations of differentiation and integration for functions on a closed interval [a, b]. The first part asserts that if f is continuous on [a, b], then the function F(x) = \int_a^x f(t) \, dt is differentiable on (a, b) with F'(x) = f(x) for all x \in (a, b), and the one-sided derivatives at the endpoints satisfy F'_+(a) = f(a) and F'_-(b) = f(b); moreover, F is continuous on [a, b]. The second part states that if F is differentiable on [a, b] (with one-sided derivatives at endpoints) with F' Riemann integrable on [a, b], then \int_a^b F'(x) \, dx = F(b) - F(a). These theorems demonstrate that differentiation and (Riemann) integration are inverse operations under appropriate conditions, providing a foundational bridge between the two concepts in real analysis. To prove the second part, consider a partition P = \{a = x_0 < x_1 < \cdots < x_n = b\} of [a, b]. By the mean value theorem applied to F on each subinterval [x_{i-1}, x_i], there exists c_i \in (x_{i-1}, x_i) such that F(x_i) - F(x_{i-1}) = F'(c_i) (x_i - x_{i-1}). Summing over i = 1 to n yields the telescoping sum F(b) - F(a) = \sum_{i=1}^n F'(c_i) (x_i - x_{i-1}), which is a Riemann sum for \int_a^b F'(x) \, dx. As the mesh of the partition approaches zero, this Riemann sum converges to the integral, so F(b) - F(a) = \int_a^b F'(x) \, dx. This proof relies on the Riemann integrability of F' and the continuity of F implied by differentiability. The integration by parts formula follows directly from the product rule for differentiation and the second part of the fundamental theorem. Suppose u and v are differentiable on [a, b] with u' and v' Riemann integrable. Let w(x) = u(x) v(x); then w'(x) = u'(x) v(x) + u(x) v'(x). Integrating both sides gives \int_a^b w'(x) \, dx = \int_a^b u'(x) v(x) \, dx + \int_a^b u(x) v'(x) \, dx. By the second part, the left side equals w(b) - w(a) = u(b) v(b) - u(a) v(a), so \int_a^b u(x) v'(x) \, dx = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx, or in standard notation, \int_a^b u \, dv = [u v]_a^b - \int_a^b v \, du. This technique is particularly useful for integrating products of functions where one derivative simplifies the expression. Similarly, the substitution rule, or change of variables formula, derives from the chain rule and the fundamental theorems. Suppose g is differentiable on [c, d] with g' Riemann integrable, g([c, d]) = [a, b], and f is continuous on [a, b]. Let F be an antiderivative of f, so F'(x) = f(x). Then F(g(x)) has derivative f(g(x)) g'(x) by the chain rule. Integrating yields \int_c^d f(g(x)) g'(x) \, dx = \int_c^d \frac{d}{dx} [F(g(x))] \, dx = F(g(d)) - F(g(c)) = F(b) - F(a) = \int_a^b f(u) \, du, where u = g(x). This rule facilitates evaluation of integrals by reversing substitutions used in differentiation. As consequences, continuous functions on [a, b] are Riemann integrable, as established earlier using uniform continuity to ensure the upper and lower Darboux integrals coincide. Additionally, the theorems ensure that the derivative of an integral recovers the integrand, enabling practical computation of integrals via antiderivatives and confirming the invertibility of these operations for suitable classes of functions. Moreover, antiderivatives of Riemann integrable functions are continuous. Historically, the intuitive formulation of these theorems emerged in the 17th century through the independent work of Isaac Newton and Gottfried Wilhelm Leibniz, who recognized the inverse relationship between "fluxions" (derivatives) and "fluents" (integrals) while developing calculus for physical applications. Rigorous proofs, based on limits and addressing foundational issues like the nature of infinitesimals, were provided by Augustin-Louis Cauchy in his 1821 textbook Cours d'analyse de l'École Royale Polytechnique, where he established the modern epsilon-delta framework for continuity and convergence underlying the theorems.

Improper integrals

Improper integrals arise in real analysis as an extension of the Riemann integral to handle functions over unbounded intervals or those with discontinuities, such as infinite discontinuities, within a finite interval. These integrals are defined using limits of proper Riemann integrals. For an unbounded interval, the improper integral \int_a^\infty f(x) \, dx is defined as \lim_{b \to \infty} \int_a^b f(x) \, dx, provided the limit exists and is finite; if the limit does not exist or is infinite, the integral diverges. Similarly, \int_{-\infty}^b f(x) \, dx = \lim_{a \to -\infty} \int_a^b f(x) \, dx, and for the entire real line, \int_{-\infty}^\infty f(x) \, dx = \lim_{a \to -\infty, b \to \infty} \int_a^b f(x) \, dx, where the double limit requires the iterated limits to agree for convergence. For singularities at a finite point c in [a, b], the integral is split as \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx, with each part defined as a limit approaching c from the appropriate side; for example, \int_0^1 \frac{1}{\sqrt{x}} \, dx = \lim_{\epsilon \to 0^+} \int_\epsilon^1 \frac{1}{\sqrt{x}} \, dx = 2. Convergence of improper integrals can be tested using several criteria analogous to those for series. The comparison test states that if $0 \leq f(x) \leq g(x) for x \geq a and \int_a^\infty g(x) \, dx converges, then \int_a^\infty f(x) \, dx converges; conversely, if \int_a^\infty f(x) \, dx diverges, then \int_a^\infty g(x) \, dx diverges. The limit comparison test applies when f(x) > 0, g(x) > 0 for large x, and \lim_{x \to \infty} \frac{f(x)}{g(x)} = L where $0 < L < \infty; in this case, \int_a^\infty f(x) \, dx and \int_a^\infty g(x) \, dx either both converge or both diverge. For oscillatory integrands, the Dirichlet test guarantees convergence of \int_a^\infty f(x) g(x) \, dx if the partial integrals \left| \int_a^x f(t) \, dt \right| are bounded for all x \geq a and g(x) is monotonic with \lim_{x \to \infty} g(x) = 0; the Abel test is a related criterion where g(x) has bounded variation instead of being monotonic. An improper integral converges absolutely if \int_a^\infty |f(x)| \, dx < \infty, which implies ordinary convergence by the comparison test with |f(x)|; however, convergence without absolute convergence is possible and termed conditional. A classic example is the Dirichlet integral \int_1^\infty \frac{\sin x}{x} \, dx, which converges conditionally by the Dirichlet test (with f(x) = \sin x and g(x) = 1/x), but \int_1^\infty \left| \frac{\sin x}{x} \right| \, dx diverges by comparison to the harmonic series, as the absolute value creates intervals of length \pi where it behaves like $1/x. Representative examples illustrate these concepts. The p-integrals \int_1^\infty x^{-p} \, dx converge if and only if p > 1, evaluating to \frac{1}{p-1} when convergent, and diverge otherwise; this serves as a benchmark for the comparison test. The Gamma function provides another key example, defined as \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt for \operatorname{Re}(z) > 0, where convergence holds due to the exponential decay dominating the power at infinity and the power being integrable near zero. Improper integrals relate closely to infinite series via the integral test: if f(x) is positive, continuous, and decreasing on [1, \infty), then the series \sum_{n=1}^\infty f(n) converges if and only if \int_1^\infty f(x) \, dx converges. This test determines the convergence of p-series \sum_{n=1}^\infty \frac{1}{n^p}, which converge precisely when p > 1, mirroring the p-integral behavior.

Measure and Lebesgue Integration

Lebesgue measure

The Lebesgue measure provides a rigorous generalization of the intuitive notion of length for subsets of the real line \mathbb{R}, extending beyond the limitations of Jordan content by assigning measures to a broader class of sets in a translation-invariant and countably additive manner. Developed by Henri Lebesgue in the early 20th century, it forms the foundation for modern integration theory, allowing the measure of sets that are not Riemann-integrable in the classical sense. The construction begins with the definition of an outer measure, which approximates the "size" of any subset from above using countable covers by intervals. The Lebesgue outer measure m^*(E) of a subset E \subseteq \mathbb{R} is defined as the infimum of the sums of lengths of countable collections of open intervals that cover E: m^*(E) = \inf \left\{ \sum_{k=1}^\infty \ell(I_k) : \{I_k\}_{k=1}^\infty \text{ is a countable cover of } E \text{ by open intervals } I_k \right\}, where \ell(I_k) denotes the length of the interval I_k. This outer measure is well-defined for all subsets of \mathbb{R}, non-negative, and assigns zero to the empty set. It satisfies monotonicity: if E \subseteq F, then m^*(E) \leq m^*(F), and subadditivity: for any countable collection \{E_k\}, m^*\left(\bigcup_k E_k\right) \leq \sum_k m^*(E_k). A set E \subseteq \mathbb{R} is Lebesgue measurable if it satisfies the Carathéodory criterion: for every set A \subseteq \mathbb{R}, m^*(A) = m^*(A \cap E) + m^*(A \setminus E). This condition, introduced by Constantin Carathéodory, ensures that measurable sets split the outer measure additively and generates a \sigma-algebra \mathcal{M} of measurable sets closed under countable unions and complements. The Lebesgue measure m is then the restriction of m^* to \mathcal{M}, so m(E) = m^*(E) for measurable E. Key properties of the Lebesgue measure include translation invariance: for any measurable E and real number c, m(E + c) = m(E), reflecting that shifting a set does not change its measure. Monotonicity holds for measurable sets: if E \subseteq F and both are measurable, then m(E) \leq m(F). Moreover, countable additivity applies: if \{E_k\}_{k=1}^\infty are pairwise disjoint measurable sets, then m\left(\bigcup_k E_k\right) = \sum_k m(E_k), enabling the measure of complicated sets via decomposition. For intervals, the measure coincides with length: m((a,b)) = b - a. The Borel \sigma-algebra \mathcal{B}(\mathbb{R}) is the smallest \sigma-algebra containing all open intervals, generated by countable unions, intersections, and complements starting from these intervals. All Borel sets are Lebesgue measurable, with m restricting to the Borel measure on \mathcal{B}(\mathbb{R}), ensuring that familiar sets like open, closed, and compact sets have well-defined measures. However, not all subsets of \mathbb{R} are measurable; the axiom of choice implies the existence of non-measurable sets. A classic example of a measurable set with measure zero is the Cantor set, constructed by iteratively removing middle-third open intervals from [0,1]. The resulting set C is uncountable (homeomorphic to \{0,1\}^\mathbb{N}) yet has Lebesgue measure m(C) = 0, as the total length removed sums to 1. This illustrates that measure zero sets can be "large" in cardinality. In contrast, the Vitali set V, constructed using the axiom of choice by selecting one representative from each equivalence class of \mathbb{R}/\mathbb{Q} in [0,1], is non-measurable: its countable disjoint translates by rationals cover [0,1] up to measure zero, but additivity would imply contradictory measures for V. While the construction focuses on \mathbb{R}, the Lebesgue measure extends to \mathbb{R}^n as the product measure of one-dimensional measures on each coordinate, preserving translation invariance and countable additivity for Borel sets in higher dimensions.

Measurable functions and integration

A measurable function is a function f: \mathbb{R} \to \mathbb{R} such that for every Borel set B \subseteq \mathbb{R}, the preimage f^{-1}(B) is a Lebesgue measurable set. This definition ensures that measurable functions preserve the structure of measurability under the Lebesgue measure, allowing integration over sets of arbitrary complexity. Simple functions form the building blocks for Lebesgue integration and are defined as finite linear combinations of indicator functions of measurable sets, typically nonnegative for initial constructions: \phi = \sum_{k=1}^n c_k \chi_{E_k}, where c_k \geq 0 and each E_k is measurable. The integral of such a simple function \phi with respect to the Lebesgue measure m is \int \phi \, dm = \sum_{k=1}^n c_k m(E_k). For a nonnegative measurable function f: \mathbb{R} \to [0, \infty], the Lebesgue integral is defined as the supremum of the integrals of simple functions approximating f from below: \int f \, dm = \sup\left\{ \int \phi \, dm : 0 \leq \phi \leq f, \, \phi \text{ simple} \right\}. This construction extends the notion of integration beyond continuous functions by weighting sets according to their Lebesgue measure. For signed measurable functions f = f^+ - f^-, where f^+ = \max(f, 0) and f^- = \max(-f, 0), the integral is \int f \, dm = \int f^+ \, dm - \int f^- \, dm, provided at least one is finite. Key convergence properties underpin the power of Lebesgue integration. The monotone convergence theorem states that if $0 \leq f_n \uparrow f pointwise, where each f_n is measurable and f is the pointwise limit, then \int f_n \, dm \uparrow \int f \, dm. (Note: This result, often associated with Lebesgue's framework, was originally proved by Beppo Levi in the context of countable additivity for series.) The dominated convergence theorem provides a condition for interchanging limits and integrals: if |f_n| \leq g for some integrable g (i.e., \int g \, dm < \infty), f_n \to f almost everywhere, and each f_n is measurable, then \int f_n \, dm \to \int f \, dm. In comparison to the Riemann integral, Lebesgue integration encompasses a broader class of functions, including many discontinuous ones that are Riemann non-integrable, while agreeing on continuous functions; for instance, \int_0^1 x \, dx = \frac{1}{2} in both theories. A classic example is the Dirichlet function d(x) = 1 if x is rational and $0 if irrational on [0,1], which is nowhere continuous and thus not Riemann integrable, but Lebesgue integrable with \int_0^1 d(x) \, dm = 0 since the rationals have measure zero.

Convergence theorems in Lebesgue integration

In Lebesgue integration, convergence theorems provide essential tools for interchanging limits and integrals, extending beyond the capabilities of Riemann integration by handling a broader class of functions and measures. These theorems rely on the structure of measurable functions and the Lebesgue integral defined over measure spaces, ensuring that pointwise convergence under suitable conditions implies convergence of integrals. Key results include the monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem, which collectively enable rigorous analysis of limits in integration theory. The monotone convergence theorem states that if \{f_n\} is a sequence of nonnegative measurable functions on a measure space (X, \mathcal{M}, \mu) such that f_n \uparrow f pointwise (i.e., $0 \leq f_1 \leq f_2 \leq \cdots and f_n(x) \to f(x) for all x \in X), then \int f_n \, d\mu \uparrow \int f \, d\mu. This theorem is fundamental for approximating integrable functions via increasing sequences of simple functions. To prove it, first consider the case where f is a simple function, say f = \sum_{k=1}^m c_k \chi_{E_k} with c_k > 0 and E_k disjoint measurable sets. For each n, define g_n = \sum_{k=1}^m \min(c_k, f_n) \chi_{E_k}; then g_n \uparrow f and \int g_n \, d\mu \leq \int f_n \, d\mu by monotonicity of the integral for simple functions. Since \int f_n \, d\mu - \int g_n \, d\mu \to 0 as n \to \infty (as f_n \uparrow f), it follows that \int f_n \, d\mu \to \int f \, d\mu. For general nonnegative measurable f, approximate f by an increasing sequence of simple functions \{\phi_j\} \uparrow f; then \{ \phi_j \wedge f_n \} \uparrow f for fixed n, so \int f_n \, d\mu = \sup_j \int (\phi_j \wedge f_n) \, d\mu. Taking n \to \infty and interchanging suprema using the countably additive property of \mu yields \lim_{n \to \infty} \int f_n \, d\mu = \sup_j \lim_{n \to \infty} \int (\phi_j \wedge f_n) \, d\mu = \sup_j \int \phi_j \, d\mu = \int f \, d\mu. Fatou's lemma provides a lower semicontinuity result for integrals: if \{f_n\} is a sequence of nonnegative measurable functions, then \int \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int f_n \, d\mu. The proof proceeds by defining g_n = \inf_{k \geq n} f_k, so \{g_n\} is increasing and g_n \uparrow \liminf_{n \to \infty} f_n. By the monotone convergence theorem, \int g_n \, d\mu \uparrow \int \liminf_{n \to \infty} f_n \, d\mu. Since g_n \leq f_k for all k \geq n, taking \inf_{k \geq n} \int f_k \, d\mu \geq \int g_n \, d\mu and then \liminf_{n \to \infty} \int f_n \, d\mu \geq \lim_{n \to \infty} \int g_n \, d\mu yields the inequality. Equality holds if \{f_n\} is uniformly integrable, but the lemma is sharp in general. The dominated convergence theorem asserts that if \{f_n\} is a sequence of measurable functions converging pointwise to f almost everywhere (a.e.), and there exists an integrable function g \geq 0 such that |f_n| \leq g \mu-a.e. for all n, then f is integrable and \int f_n \, d\mu \to \int f \, d\mu. To prove it, without loss of generality assume f_n \geq 0 and f \geq 0 by considering positive and negative parts separately. Define h_n = g - |f_n - f| \geq 0; then h_n \uparrow h = g - |f - f| = g a.e., so by the monotone convergence theorem, \int h_n \, d\mu \to \int g \, d\mu. But \int h_n \, d\mu = \int g \, d\mu - \int |f_n - f| \, d\mu, so \int |f_n - f| \, d\mu \to 0. Thus, | \int (f_n - f) \, d\mu | \leq \int |f_n - f| \, d\mu \to 0, and integrability of f follows from |f| \leq g. The proof uses the domination to bound the remainder and applies monotone convergence to the difference. These theorems hold under almost everywhere convergence, meaning pointwise convergence on a set of full measure (i.e., except on a set of measure zero), as modifying functions on null sets does not affect the Lebesgue integral due to the properties of measurable functions. Egorov's theorem strengthens this by providing uniform convergence on large subsets: if \{f_n\} converges pointwise a.e. to f on a set E with \mu(E) < \infty, then for every \varepsilon > 0, there exists a measurable subset F \subset E with \mu(E \setminus F) < \varepsilon such that f_n \to f uniformly on F. The proof is constructive: for each k, cover the set where \sup_{m,n \geq k} |f_m - f_n| \geq 1/j by countably many sets of measure less than \varepsilon 2^{-k}, and union over k to find a bad set of measure less than \varepsilon; the complement yields uniform Cauchy convergence, hence uniform convergence to f. This is particularly useful on finite measure spaces like probability spaces. Applications of these theorems abound, notably in interchanging limits and integrals under domination, which justifies \mathbb{E}[\lim X_n] = \lim \mathbb{E}[X_n] for random variables X_n \to X a.e. with |X_n| \leq Y integrable, a cornerstone of probability theory. For instance, in computing expectations of indicators or bounded approximations, the dominated convergence theorem ensures the limit passes inside the integral without altering the value.

Advanced Topics

Distributions and generalized functions

In real analysis, distributions extend the notion of functions by treating them as continuous linear functionals on appropriate spaces of test functions, enabling the handling of singularities and generalized derivatives in a rigorous manner. This framework, introduced by Laurent Schwartz, allows for the formulation of differential equations in weak senses and facilitates applications in partial differential equations and Fourier analysis. The space of test functions, denoted C_c^\infty(\mathbb{R}), consists of all infinitely differentiable functions on \mathbb{R} with compact support. These functions are equipped with the inductive limit topology, where convergence is defined by uniform convergence on compact sets for the functions themselves and all their derivatives of any order. A distribution T is then a linear functional T: C_c^\infty(\mathbb{R}) \to \mathbb{R} that is continuous with respect to this topology, meaning that if a sequence of test functions converges in this sense, then T applied to them converges in \mathbb{R}. Regular distributions correspond to those induced by locally integrable functions f \in L^1_{\mathrm{loc}}(\mathbb{R}), defined by \langle T_f, \phi \rangle = \int_{\mathbb{R}} f(x) \phi(x) \, dx for every test function \phi, with the integral understood in the Lebesgue sense. A canonical example of a singular distribution is the Dirac delta \delta, defined by \langle \delta, \phi \rangle = \phi(0) for all \phi \in C_c^\infty(\mathbb{R}); this cannot be realized as integration against a classical function due to its concentration at the origin. Distributions admit a notion of differentiation: the derivative T' of a distribution T satisfies \langle T', \phi \rangle = -\langle T, \phi' \rangle, which extends the classical integration-by-parts formula. For the Dirac delta, the first derivative is \langle \delta', \phi \rangle = -\phi'(0). Weak derivatives generalize this further: a locally integrable function f is said to have weak derivative g if \int_{\mathbb{R}} f(x) \phi'(x) \, dx = -\int_{\mathbb{R}} g(x) \phi(x) \, dx holds for every test function \phi, allowing derivatives to exist in a distributional sense even when classical derivatives do not. To accommodate functions with slower decay at infinity, such as those relevant for Fourier transforms, tempered distributions are defined on the Schwartz space \mathcal{S}(\mathbb{R}), which comprises all infinitely differentiable functions whose derivatives decay faster than any polynomial at infinity—formally, \phi \in \mathcal{S}(\mathbb{R}) if \sup_{x \in \mathbb{R}} |x|^k |\partial^\alpha \phi(x)| < \infty for all integers k \geq 0 and multi-indices \alpha. The topology on \mathcal{S}(\mathbb{R}) is given by seminorms involving these suprema, and a tempered distribution is a continuous linear functional on this space. Notable examples include the Heaviside step function H(x), whose distributional derivative is the Dirac delta \delta, since \langle H', \phi \rangle = -\int_0^\infty \phi'(x) \, dx = \phi(0) = \langle \delta, \phi \rangle. Another is the Cauchy principal value distribution associated with $1/x, defined for odd test functions by \langle \mathrm{p.v.} \, 1/x, \phi \rangle = \lim_{\epsilon \to 0^+} \int_{|x| > \epsilon} \frac{\phi(x)}{x} \, dx, which extends the singular function $1/x to a tempered distribution.

Relation to complex analysis

The complex numbers \mathbb{C} can be constructed as the Euclidean plane \mathbb{R}^2 equipped with a field structure, where addition is componentwise and multiplication is defined by (x_1, y_1) \cdot (x_2, y_2) = (x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2), making \mathbb{C} an algebraically closed field extension of \mathbb{R}. This identification allows real analysis tools, such as partial derivatives, to be applied to functions on \mathbb{C}. A function f: D \to \mathbb{C}, where D \subset \mathbb{C} is open, is holomorphic if it is complex differentiable at every point in D, meaning the limit \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} exists for all z_0 \in D; such functions are analytic, expressible as power series converging uniformly on compact subsets of D. Writing f(z) = u(x,y) + i v(x,y) with z = x + i y and u, v: \mathbb{R}^2 \to \mathbb{R}, holomorphy is equivalent to u and v satisfying the Cauchy-Riemann equations \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} at every point, assuming the partial derivatives exist and are continuous; these equations link the real partial derivatives to the existence of the complex derivative f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}. If f is holomorphic in a simply connected domain without singularities, Cauchy's theorem states that the contour integral \oint_\gamma f(z) \, dz = 0 for any closed curve \gamma in the domain, extending real line integrals to paths in the complex plane and relying on the real fundamental theorem of calculus for path independence. The residue theorem generalizes this: for a closed contour \gamma enclosing isolated singularities of f, \oint_\gamma f(z) \, dz = 2\pi i \sum \operatorname{Res}(f, z_k), where the sum is over residues at poles z_k inside \gamma, enabling evaluation of real improper integrals by closing contours in the complex plane. For example, the integral \int_{-\infty}^\infty \frac{1}{1+x^2} \, dx = \pi can be computed by considering the contour integral of \frac{1}{1+z^2} over a semicircular contour in the upper half-plane; the simple pole inside is at z = i with residue \frac{1}{2i}, so the integral is $2\pi i \times \frac{1}{2i} = \pi, and the contribution from the arc vanishes as the radius tends to infinity. Analytic continuation extends real functions defined on subsets of \mathbb{R} to holomorphic functions on larger domains in \mathbb{C}, uniquely determined by their values on any set with a limit point; for instance, the real rational function \frac{1}{1+x^2} extends to f(z) = \frac{1}{1+z^2} = \frac{1}{(z-i)(z+i)} on \mathbb{C} \setminus \{\pm i\}, revealing simple poles at z = \pm i and allowing computation of real integrals via residues at these points. Unlike real analysis, where convergence of sequences of functions is pointwise or uniform without strong global bounds, complex analysis benefits from the maximum modulus principle: if f is holomorphic and non-constant in a bounded domain D and continuous up to the boundary, then \max_{z \in \overline{D}} |f(z)| = \max_{z \in \partial D} |f(z)|, implying uniform convergence on compact subsets for power series and preventing interior maxima, which strengthens results like Liouville's theorem on bounded entire functions.

Generalizations to metric spaces

Many concepts from real analysis on the real line, such as limits, continuity, and compactness, generalize directly to the broader framework of metric spaces, where the real numbers serve as the prototypical example with the standard absolute value metric. A metric space is a set X equipped with a metric d: X \times X \to [0, \infty) that satisfies: d(x, y) = 0 if and only if x = y; d(x, y) = d(y, x) for all x, y \in X; and the triangle inequality d(x, z) \leq d(x, y) + d(y, z) for all x, y, z \in X. In such a space, a sequence (x_n)_{n=1}^\infty in X converges to a point x \in X if for every \epsilon > 0, there exists N \in \mathbb{N} such that d(x_n, x) < \epsilon for all n > N. The sequence is Cauchy if for every \epsilon > 0, there exists N \in \mathbb{N} such that d(x_m, x_n) < \epsilon for all m, n > N. The metric space (X, d) is complete if every Cauchy sequence converges to some point in X. The real line \mathbb{R} with d(x, y) = |x - y| is a complete metric space, as every Cauchy sequence of reals converges to a real limit. However, subspaces like the rational numbers \mathbb{Q} with the same metric are incomplete, since sequences of rationals can be Cauchy yet converge to irrational limits outside \mathbb{Q}. A function f: (X, d_X) \to (Y, d_Y) between metric spaces is continuous at x \in X if for every \epsilon > 0, there exists \delta > 0 such that d_X(x', x) < \delta implies d_Y(f(x'), f(x)) < \epsilon for all x' \in X; this \epsilon-\delta definition mirrors that on the reals. The function is uniformly continuous if for every \epsilon > 0, there exists \delta > 0 such that d_X(x', x'') < \delta implies d_Y(f(x'), f(x'')) < \epsilon for all x', x'' \in X, with \delta independent of the points. In metric spaces, a subset is compact if every open cover has a finite subcover; equivalently, every sequence has a convergent subsequence (sequential compactness). While the Heine-Borel theorem characterizes compact subsets of \mathbb{R}^n as precisely the closed and bounded ones, this fails in infinite-dimensional metric spaces. For instance, the closed unit ball in the Hilbert space \ell^2 (sequences of squares-summable reals with the \ell^2-metric) is closed and bounded but not compact, as it contains sequences without convergent subsequences. Complete metric spaces are Baire spaces: they cannot be expressed as a countable union of nowhere dense sets (meager sets), meaning the complement of any meager set is dense, and countable intersections of dense open sets are dense. This theorem, originally due to René Baire, has applications in real analysis; for example, in the space C[0,1] of continuous functions on [0,1] with the supremum metric, the set of functions differentiable at least at one point is meager, so "most" continuous functions are nowhere differentiable. A Polish space is a separable complete metric space (or a space homeomorphic to one), where separability means a countable dense subset exists. The real line \mathbb{R} is a canonical Polish space, and these spaces provide a foundation for extending measure theory beyond \mathbb{R}^n, as their topology supports Borel \sigma-algebras. The space C[0,1] of continuous real-valued functions on [0,1] equipped with the supremum metric d(f, g) = \sup_{x \in [0,1]} |f(x) - g(x)| is complete, since uniform limits of continuous functions are continuous, but it is not compact due to its infinite dimensionality—sequences of functions like f_n(x) = x^n have no convergent subsequence in the metric.

References

  1. [1]
    [PDF] Introduction to Real Analysis
    Jun 8, 2021 · This first volume is a one semester course in basic analysis. Together with the second volume it is a year-long course. It started its life as ...
  2. [2]
    [PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
    Abstract. These are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits.
  3. [3]
    [PDF] MATH 321-1: Real Analysis - Northwestern University, Lecture Notes
    Real analysis is the study of functions defined on the set of real numbers, or subsets thereof. The key concepts we care about this quarter are continuity ...
  4. [4]
    [PDF] Stetigkeit und irrationale Zahlen. - ETH Zurich
    Stetigkeit und irrationale Zahlen. Von. Richard Dedekind,. Professor der Mathematik an der technischen Hochschule zu Braunschweig. Nach der zweiten ...
  5. [5]
    https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/10%3A_Epilogue_to_Real_Analysis/10.02%3A_Building_the_Real_Numbers
  6. [6]
    [PDF] Ueber die Ausdehnung eines Satzes aus der Theorie der ...
    Ueber die Ausdehnung eines Satzes aus der. Theorie der trigonometrischen Reihen. Von G. Cantor in Halle a. S. [Math. Annalen 5, 123–132 (1872).] Im folgenden ...
  7. [7]
    [PDF] Math 117: Axioms for the Real Numbers
    Oct 11, 2010 · A complete ordered field is an ordered field F such that if a nonempty subset S ⊂ F has an upper bound, then S has a least upper bound or ...
  8. [8]
    [PDF] Supplement. The Real Numbers are the Unique Complete Ordered ...
    Oct 2, 2024 · In this supplement we prove that, up to isomorphism, there is only one complete ordered field. The source for this supplement is Michael Henle's ...
  9. [9]
    1.4: Ordered Field Axioms - Mathematics LibreTexts
    Sep 5, 2021 · + and · ⋅ and a relation · < satisfying the 13 axioms above is called an ordered field. Thus the real numbers are an example of an ordered field.
  10. [10]
    [PDF] Math 117: Density of Q in R
    Theorem (Q is dense in R). For every x, y ∈ R such that x<y, there exists a rational number r such that x<r<y.
  11. [11]
    [PDF] 1. The AM-GM inequality - Berkeley Math Circle
    It follows that if x, y ≥ 0 and x 6= y, then inequality is strict: (x + y)/2 >. √ xy. Here's a one-line proof of the AM-GM inequality for two variables: x + y.
  12. [12]
    [PDF] Real Numbers 1 Axioms of the Real Numbers
    Real numbers have field axioms (like rational numbers) and the least upper bound property, which is also called the (order) completeness of R.
  13. [13]
    [PDF] completeness of the real numbers - UTK Math
    In any ordered field F, the Supremum Property and the. Monotone Convergence Property are equivalent; either of them implies the. Archimedean Property. Example ...
  14. [14]
    [PDF] Chapter 3. Topology of the Real Numbers.
    Aug 12, 2023 · Open intervals are examples of open sets, but there are open sets that are not intervals. For example, the union of two disjoint open intervals ...
  15. [15]
    [PDF] Topology of the Real Numbers - UC Davis Math
    This chapter defines topological properties of real numbers, including open sets, which are defined as sets where every point has a neighborhood within the set.
  16. [16]
    3 Sequences - Real Analysis
    Many concepts in analysis can be described using the long-term or limiting behavior of sequences. In calculus, you undoubtedly developed techniques to compute ...Limit Theorems · Bolzano-Weierstrass Theorem · Cauchy Sequences
  17. [17]
    6.5 (Optional) Properties of limits
    🔗. The limit of a sequence is unique · 🔗. Linearity of limits: . lim n → ∞ ( c ⋅ x n + d ⋅ y n ) = c ⋅ a + d ⋅ b . · 🔗. Product of limits: . · 🔗. Reciprocal of ...
  18. [18]
    2.1 Sequences and limits
    Analysis is essentially about taking limits. The most basic type of a limit is a limit of a sequence of real numbers.
  19. [19]
    [PDF] 3.6 Cauchy Sequences
    A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Here is the formal definition. Definition 3.6.
  20. [20]
    Cauchy Sequences
    A new type of sequence in which the terms are not said to grow arbitrarily close to a fixed limit, but instead arbitrarily close to each other.
  21. [21]
    [PDF] 18.100A Fall 2020 Lecture 10: The Completeness of the Real ...
    Theorem 7. A sequence of real numbers {xn} is Cauchy if and only if {xn} is convergent. Proof: ( =⇒ ) If {xn} is Cauchy, then {xn} is bounded. Therefore, {xn} ...
  22. [22]
    [PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
    These are some notes on introductory real analysis. They cover limits of functions, continuity, differentiability, and sequences and series of functions, but ...
  23. [23]
    [PDF] 1.6 The Nested Intervals Theorem
    The advantage of working with Cauchy sequences is that it gives a condition of convergence of a sequence without specifying what the sequence converges to. And ...
  24. [24]
    [PDF] Cauchy's Construction of R - UCSD Math
    Theorem 2.5. If (an) is a Cauchy sequence, then it is bounded; that is, there is some large number M such that |an| ≤ M for all n.
  25. [25]
    [PDF] The Real Numbers. - UCSD Math
    Example 1. A Sequence of Rational Numbers Approaching √2 found by Newton's Method. First recall Newton's method for approximating roots of equations f(x)=0. у ...
  26. [26]
    [PDF] Section 9.4. Complete Metric Spaces
    Apr 25, 2023 · A metric space X is complete if every Cauchy sequence in X converges to a point in X. Note. The real numbers form a metric space with the usual ...
  27. [27]
    Complete Metric Spaces - Advanced Analysis
    Jan 17, 2024 · A metric space ( X , d ) is called complete when all Cauchy sequences in the metric space are convergent. Examples. The real numbers with the ...
  28. [28]
    1.2: Epsilon-Delta Definition of a Limit - Mathematics LibreTexts
    Dec 20, 2020 · This section introduces the formal definition of a limit. Many refer to this as "the epsilon--delta,'' definition, referring to the letters ...
  29. [29]
    [PDF] Functional Limits and Continuity
    Sequential Criterion for Functional Limits. Functional limits can be completely char- acterized by the convergence of all related sequences. Theorem 4.1 ( ...<|control11|><|separator|>
  30. [30]
    2.7: The Precise Definition of a Limit - Mathematics LibreTexts
    Dec 20, 2020 · In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language.
  31. [31]
    https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/03:_Limits_and_Continuity/3.02:_Definition_of_Continuity
  32. [32]
    https://math.libretexts.org/Bookshelves/Calculus/CLP-1_Differential_Calculus_(Feldman_Rechnitzer_and_Yeager)/02:_Limits/2.06:_Continuity
  33. [33]
    https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/04:_Function_Limits_and_Continuity/4.09:_The_Intermediate_Value_Property
  34. [34]
    [PDF] Real-valued Functions: Continuity & Uniform Continuity
    Jul 5, 2011 · Let f be uniformly continuous on I. Then {xn} is Cauchy in I ⇒ {f(xn)} is Cauchy in R. INTERMEDIATE VALUE THEOREM: • Let f be continuous on I.
  35. [35]
    Continuity | An Introduction to Real Analysis
    The function f is continuous at c ∈ A if for any given ε > 0 there exists δ > 0 such that if x ∈ A and | x − c | < δ then | f ( x ) − f ( c ) | < ε .
  36. [36]
    [PDF] Real Analysis MAA 6616 Lecture 22 Absolutely Continuous Functions
    Real Analysis MAA 6616. Lecture 22. Absolutely ... bj − aj. Therefore for > 0, we can take δ = /c for f to satisfy the definition of absolute continuity.
  37. [37]
    Absolute Continuity
    Jan 8, 2018 · We can apply our result that absolutely continuous functions are also of bounded variation and, hence, are the difference of two increasing ...
  38. [38]
    [PDF] A simple proof of the Fundamental Theorem of Calculus for ... - arXiv
    Mar 7, 2012 · the main connection between absolute continuity and Lebesgue integration: ... Botsko, The use of full covers in real analysis, Amer. Math ...<|control11|><|separator|>
  39. [39]
    [PDF] Shape Analysis, Lebesgue Integration and Absolute Continuity ...
    Jul 7, 2018 · We have reviewed fundamental concepts and results about Lebesgue integra- tion and absolute continuity, some results connecting the two notions, ...
  40. [40]
    Absolutely Continuous Functions
    Note that the Cantor function does not satisfy the (N) property since it sends a set of Lebesgue measure zero, the Cantor set D, into the full interval [0,1].
  41. [41]
    [PDF] arXiv:1406.0204v1 [math.DS] 1 Jun 2014
    Jun 1, 2014 · and easy to see that νλ is singular, as it is supported on a Cantor set of Hausdorff ... is absolutely continuous (even with an L2 density) for ...
  42. [42]
    [PDF] arXiv:1810.05916v2 [math.CV] 25 Feb 2021
    Feb 25, 2021 · The property of absolute continuity in measure is also called Lusin's condition (N) in the literature. An important fact is that any ...
  43. [43]
    [PDF] 1.7 The Heine-Borel Covering Theorem; open sets, compact sets
    The Heine-Borel theorem says that closed bounded intervals [a, b] are examples of compact sets. The concept of open set is what is needed in order to define ...
  44. [44]
    [PDF] 16. Compactness
    Theorem 4.2 (Heine-Borel theorem for R). A subset of Rusual is compact if and only if it is closed and bounded. Proof. (⇒). Suppose K ⊆ R ...
  45. [45]
    [PDF] Section 26. Compact Sets
    Jul 27, 2016 · Recall that, in the real setting, a continuous function on a compact set attains a maximum and minimum (the Extreme Value. Theorem) and a ...
  46. [46]
    [PDF] COMPACTNESS VS. SEQUENTIAL - MIT OpenCourseWare
    The aim of this handout is to provide a detailed proof of the equivalence between the two definitions of compactness: existence of a finite subcover of any ...
  47. [47]
    5.2. Compact and Perfect Sets - Real Analysis - MathCS.org
    A set S of real numbers is called compact if every sequence in S has a subsequence that converges to an element again contained in S. Examples 5.2. 2:
  48. [48]
    [PDF] Open Covers and Compactness
    Closed subsets of compact sets are compact. If F is closed and K is compact then F ∩ K is compact. If {Kα : α ∈ I} is a collection of compact subsets of a ...
  49. [49]
    [PDF] Connected Sets
    Theorem 5. A subset of R is connected if and only if it is an interval. Proof. Suppose that C is a connected subset of R.
  50. [50]
    [PDF] II.2. Connectedness
    Dec 6, 2023 · A set A is connected if there is no separation of the set. In R, the connected sets are intervals and singletons (this is Proposition II.2.2).
  51. [51]
    [PDF] CONNECTED Sets and the Intermediate VALUE THEOREM
    The most important example of a connected space is an interval in R, which means either an open interval, closed interval, or half-open interval. The limits can ...
  52. [52]
    [PDF] 5. Connectedness
    Connectedness and path-connectedness are not equivalent. We saw that the ... Thus, the connected subspaces of R are path-connected. As is the case for ...
  53. [53]
    [PDF] THE CANTOR SET
    With this definition we can prove two more important facts about the Cantor set. Theorem 1.7. The Cantor set C is perfect and totally disconnected. Proof ...
  54. [54]
    Real Analysis: Theorem 6.5.9: Mean Value Theorem - MathCS.org
    Theorem 6.5.9: Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that f'(c) =
  55. [55]
    Calculus I - The Mean Value Theorem - Pauls Online Math Notes
    Nov 16, 2022 · In this section we will give Rolle's Theorem and the Mean Value Theorem. With the Mean Value Theorem we will prove a couple of very nice ...
  56. [56]
    [PDF] PROOF OF L'HÔPITAL'S RULE - Macmillan Learning
    The proof of L'Hôpital's Rule makes use of the following generalization of the Mean. Value Theorem known as Cauchy's Mean Value Theorem. THEOREM 2 Cauchy's Mean ...
  57. [57]
    [PDF] Some consequences of the mean value theorem
    If in addition a function is twice differentiable on an interval, then the sign of the second derivative actually tells us about the convexity of a function, ...
  58. [58]
    proof of Bernoulli's inequality employing the mean value theorem
    Mar 22, 2013 · proof of Bernoulli's inequality employing the mean value theorem ... This can be proved using exactly the same method, by fixing α α in ...
  59. [59]
    example of differentiable function which is not continuously ...
    Mar 22, 2013 · f(x) = x^2sin(1/x) if x≠0, 0 if x=0. f' is not continuous because lim f'(x) as x approaches 0 diverges. f is differentiable but not C1.<|control11|><|separator|>
  60. [60]
    Taylor series expansions - MyWeb
    Sep 6, 2022 · Lagrange form​​ Theorem (Taylor): If f ( n + 1 ) exists over an open interval containing ( x , x 0 ) , then there exists x ¯ ∈ ( x , x 0 ) : R n ...
  61. [61]
    [PDF] Taylor's Series of sin x - MIT OpenCourseWare
    Taylor's Series of sin x. In order to use Taylor's formula to find the power series expansion of sin x we have to compute the derivatives of sin(x): sin (x) = ...
  62. [62]
    When Functions Are Equal to Their Taylor Series
    We can see by this that a function is equal to its Taylor series if its remainder converges to 0; i.e., if a function f can be differentiated infinitely many ...
  63. [63]
    [PDF] Absolute Convergence and the Comparison Test for Series
    Theorem 10 (Comparison Test). Suppose for all n ∈ N 0 ≤ xn ≤ yn. Then,. 1. if P yn converges, then P xn converges. 2. if P xn diverges, then P yn diverges.
  64. [64]
    https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec111.pdf
  65. [65]
    Ratio Test -- from Wolfram MathWorld
    1. If rho<1 , the series converges. 2. If rho>1 or rho=infty , the series diverges. 3. If rho=1 , the series may converge or diverge.
  66. [66]
    Leibniz Criterion -- from Wolfram MathWorld
    Also known as the alternating series test. Given a series sum_(n=1)^infty(-1)^(n+1)a_n with a_n>0, if a_n is monotonic decreasing as n->infty and ...Missing: original paper
  67. [67]
    Harmonic Series -- from Wolfram MathWorld
    The series sum_(k=1)^infty1/k is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function 1/x.
  68. [68]
    [PDF] Chapter 10: Power Series - UC Davis Math
    The basic facts are these: Every power series has a radius of convergence 0 ≤ R ≤ ∞, which depends on the coefficients an. The power series converges ...
  69. [69]
    8.3. Series and Power Series - Real Analysis - MathCS.org
    Note that it is possible for the radius of convergence to be zero (i.e. the power series converges only for x = c) or to be (i.e. the series converges for all x) ...
  70. [70]
    Fourier Series -- from Wolfram MathWorld
    A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality ...
  71. [71]
    https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/08%3A_Back_to_Power_Series/8.03%3A_Radius_of_Convergence_of_a_Power_Series
  72. [72]
    [PDF] Unit 30: Dirichlet's Proof - Harvard Mathematics Department
    It is a theorem due to Peter Gustav. Dirichlet from 1829. Theorem: The Fourier series of f ∈ X converges at every point of continuity. At discontinuities, it ...
  73. [73]
    [PDF] 18.085: Fourier Series - MIT Mathematics
    Some years later, his student, Peter Dirichlet, gave for the first time sufficient conditions on a function f(x) under which the Fourier series converges. This ...
  74. [74]
    https://people.math.harvard.edu/~knill/teaching/math22b2019/handouts/lecture30.pdf
  75. [75]
    Parseval's Theorem -- from Wolfram MathWorld
    If a function has a Fourier series given by f(x)=1/2a_0+sum_(n=1)^inftya_ncos(nx)+sum_(n=1)^inftyb_nsin(nx), (1) then Bessel's inequality becomes an ...
  76. [76]
    Fourier Series--Square Wave -- from Wolfram MathWorld
    Consider a square wave f(x) of length 2L. Over the range [0,2L], this can be written as f(x)=2[H(x/L)-H(x/L-1)]-1, (1) where H(x) is the Heaviside step ...Missing: definition real
  77. [77]
    [PDF] The Riemann Integral - UC Davis Math
    so the Riemann sums are “squeezed” between the upper and lower sums. The following theorem shows that the Darboux and Riemann definitions lead to the same ...
  78. [78]
    [PDF] Chapter 5. Integration §1. The Riemann Integral Let a and b be two ...
    If the set of discontinuities of f is finite, then f is integrable on [a, b]. Proof. Let D be the set of discontinuities of f. By our assumption, D is finite.<|separator|>
  79. [79]
    [PDF] Introduction to real analysis - William F. Trench
    Jan 2, 2016 · The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the ...
  80. [80]
    Calculus history - MacTutor - University of St Andrews
    The main ideas which underpin the calculus developed over a very long period of time indeed. The first steps were taken by Greek mathematicians.
  81. [81]
    Augustin-Louis Cauchy (1789 - 1857) - Biography - MacTutor
    Augustin-Louis Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation groups. He also researched in convergence and ...
  82. [82]
    [PDF] Basic Analysis: Introduction to Real Analysis
    May 29, 2013 · These integrals are called improper integrals, and are limits of integrals rather than integrals themselves. Definition 5.5.1. Suppose f ...
  83. [83]
    https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/
  84. [84]
    Improper integral $\sin(x)/x $ converges absolutely, conditionally or ...
    May 13, 2013 · Improper integral of sin(x)/x converges absolutely, conditionally or diverges? ... So ∫∞1sinxxdx converges. Now I need to find out if ∫∞1|sinxx|dx ...Proving $f(x) = \frac{\sin x}{x}$ converges by improper integral test?Show that the improper integral is conditionally convergentMore results from math.stackexchange.com
  85. [85]
    Gamma Function -- from Wolfram MathWorld
    [Gamma(z)]^n . The gamma function can be defined as a definite integral for R[z]>0 (Euler's integral form). Gamma(z), = int_0^inftyt^(z-1)e^(-t)dt. (3).
  86. [86]
    Calculus II - Integral Test - Pauls Online Math Notes
    Nov 16, 2022 · In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. The Integral Test can be ...
  87. [87]
    Leçons sur l'intégration et la recherche des fonctions primitives ...
    Mar 28, 2006 · Leçons sur l'intégration et la recherche des fonctions primitives, professées au Collège de France. by: Lebesgue, Henri Léon, 1875-1941.
  88. [88]
    [PDF] lebesgue measure
    Aug 28, 2024 · Outer Measure. We begin our construction of outer measure by examining the lengths of intervals. For any nonempty interval of real numbers, I ...
  89. [89]
    Über das lineare Maß von Punktmengen - EuDML
    Carathéodory, C.. "Über das lineare Maß von Punktmengen- eine Verallgemeinerung des Längenbegriffs." Nachrichten von der Gesellschaft der Wissenschaften zu ...Missing: Constantin | Show results with:Constantin
  90. [90]
    [PDF] 1.5 Carathéodory's Criterion for Lebesgue Measurability
    In contrast, the statement of Carathéodory Criterion for measur- ability only involves the definition of exterior Lebesgue measure. This makes this ...
  91. [91]
    [PDF] Chapter 2: Lebesgue Measure - UC Davis Math
    The construction, due to Carathéodory, works for any outer measure, as given in Definition 1.2, so we temporarily consider general outer measures. We will ...
  92. [92]
    [PDF] Lebesgue Outer Measure and Lebesgue Measure.
    m is translation invariant. m is countably additive. Remark 0.1 Countable additivity is important because it implies that m is (1) monotonic and (2) that m(∅) ...
  93. [93]
    [PDF] Lebesgue Measure and The Cantor Set - UNM Math
    Construct the Cantor set. • Find the measure of the Cantor set. • Show the Cantor Set is Uncountable. 2 Measure.
  94. [94]
    [PDF] Chapter 5. Product Measures - UC Davis Math
    R n are equipped with Lebesgue measure defined on their Borel σ-algebras, then the. Carathéodory σ-algebra on the product Rm+n = Rm ×Rn is the Lebesgue σ- ...
  95. [95]
    [PDF] Intégrale, Longueur, aire - Internet Archive
    Intégrale, Longueur, Aire. 2e THÈSE. — Propositions ...
  96. [96]
    [PDF] Integration - UC Davis Mathematics
    To show the linearity, we will first derive one of the fundamental convergence theorem for the Lebesgue integral, the monotone convergence theorem.Missing: source | Show results with:source
  97. [97]
    [PDF] Lecture 3 The Lebesgue Integral
    Sep 28, 2013 · Definition 3.1 (Simple functions). A function f ∈ L0(S, S, µ) is said to be simple if it takes only a finite number of values.
  98. [98]
    [PDF] 2 Lebesgue integration
    A simple example of a function not Riemann integrable is the Dirichlet function. ... Thus the Dirichlet function is Lebesgue integrable but not Riemann.<|control11|><|separator|>
  99. [99]
    [PDF] Chapter 5 Lebesgue's convergence theorems and Lp spaces
    In this chapter we study two important convergence theorems and some of their uses and applications. 5.1 Convergence theorems. Theorem 5.1. (Lebesgue's Monotone ...
  100. [100]
    [PDF] Proofs-4-3.pdf - Real Analysis
    Lebesgue Integration. 4.3. The Lebesgue Integral of a Measurable ... Fatou's Lemma. Let {fn} be a sequence of nonnegative measurable functions on ...
  101. [101]
    Introduction to Functional Analysis - MIT OpenCourseWare
    Description: We define the class of Lebesgue integrable functions and the Lebesgue integral, and we prove the powerful Dominated Convergence Theorem!
  102. [102]
    [PDF] LECTURE 13 Egoroff 's theorem (pointwise convergence is nearly ...
    The theorem is not necessarily true if µ(X) = ∞. For example, if µ is Lebesgue measure on R and fn = χ[n,n+1]. Then fn → 0 pointwise,.
  103. [103]
    [PDF] Lecture Notes in Real Analysis - University of Texas at Austin
    Dec 8, 2014 · Exercise 26. Prove Egorov's Theorem. Hint: apply the previous exercise repeatedly with δn = 2−n . Proof. By the previous exercise, there ...<|control11|><|separator|>
  104. [104]
    [PDF] Lecture 7: Interchange of integration and limit
    The proof of the main result is technical and out of the scope of this course. Theorem 2.4.2 (Lebesgue's dominated convergence theorem). Suppose that the ...
  105. [105]
    [PDF] A concise course in complex analysis and Riemann surfaces ...
    The field C of complex numbers is obtained by adjoining i to the field R of reals. The defining property of i is i2 + 1 = 0 and complex numbers z1 = x1 +iy1 and ...
  106. [106]
    [PDF] The Cauchy–Riemann Equations - UBC Math
    Jan 19, 2012 · Theorem 2 says that it is necessary for u(x, y) and v(x, y) to obey the Cauchy–Riemann equations in order for f(x + iy) = u(x + iy) + v(x + iy) ...
  107. [107]
    Mémoire sur les intégrales définies, prises entre des limites ...
    Nov 26, 2009 · Mémoire sur les intégrales définies, prises entre des limites imaginaires. by: Cauchy, Augustin Louis, Baron, 1789-1857. Publication date: 1825.
  108. [108]
    [PDF] Mémoire sur les intégrales définies, prises entre des ... - Numdam
    MÉMOIRE SUR LES INTÉGRALES DÉFINIES, PRISES ENTRE DES LIMITES IMAGINAIRES (*);. PAR M. A.-L. CAUCHY. 1 . Dans un Mémoire présenté à l'Académie des Sciences le ...
  109. [109]
    [PDF] THE GAUSSIAN INTEGRAL Let I = ∫ ∞ e dx, J ... - Keith Conrad
    Tenth Proof: Residue theorem. We will calculate. ∫ ∞. −∞ e. −x2/2 dx using contour integrals and the residue theorem. However, we can't just integrate e ...
  110. [110]
    [PDF] Chapter 3: The maximum modulus principle
    Dec 3, 2003 · By Theorem 3.6, |f(a)|≥|f(z)|∀z ∈ D(a, δ) implies f constant on D(a, δ) (since f must be analytic on D(a, δ) ⊂ G and D(a, δ) is connected open).