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Bounded function

In mathematics, particularly in real and complex analysis, a bounded function is one whose range is contained within a finite interval, meaning there exists a real number M > 0 such that |f(x)| \leq M for all x in the domain of f.^[1] This property can be refined further: a function is bounded above if there exists M such that f(x) \leq M for all x in the domain, and bounded below if there exists m such that f(x) \geq m for all x; a function is bounded if it satisfies both conditions simultaneously.^[2] Bounded functions play a central role in several foundational theorems of analysis. For instance, the Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it attains both a maximum and a minimum value on that interval, implying the function is bounded.^[3] Similarly, in the context of integration, the Riemann integral is defined only for bounded functions on closed intervals, as unbounded functions can lead to improper integrals or divergences that require separate treatment.^[4] Examples of bounded functions include constant functions and trigonometric functions like \sin x and \cos x, whose values oscillate between -1 and 1, whereas functions like f(x) = x on the real line or f(x) = 1/x on (0,1] are unbounded.^[1] The concept extends to more advanced settings, such as families of functions that are uniformly bounded—meaning a single M works for all functions in the family—or locally bounded functions, where boundedness holds in neighborhoods of each point.^[5] These notions are crucial in functional analysis, convergence theorems like the Arzelà-Ascoli theorem, and the study of metric spaces, where (by the Heine–Borel theorem) closed and bounded sets are compact in Euclidean spaces.^[6]

Definition

Real-valued functions

A function f: D \to \mathbb{R}, where D \subseteq \mathbb{R}, is said to be bounded if there exists some M > 0 such that |f(x)| \leq M for all x \in D. This condition ensures that the range of f lies within a finite interval symmetric about the origin. Formally, this can be expressed as \exists M \in \mathbb{R}^+ \ \forall x \in D, |f(x)| \leq M. A function satisfying this property is globally bounded on its domain D.^[7] Equivalently, f is bounded if and only if it is both bounded above and bounded below. The function f is bounded above on D if there exists some real number K such that f(x) \leq K for all x \in D, or equivalently, if \sup \{ f(x) \mid x \in D \} < \infty. Similarly, f is bounded below on D if there exists some real number L such that f(x) \geq L for all x \in D, or equivalently, if \inf \{ f(x) \mid x \in D \} > -\infty. This definition presupposes a basic understanding of the supremum and infimum as the least upper bound and greatest lower bound of subsets of the real numbers, respectively.^[8] While local boundedness—where the function is bounded on every compact subset of the domain—is a related concept, it is distinct from global boundedness and pertains more to analytic properties.

Functions on general domains

In the context of metric spaces, the notion of a bounded function extends the real-valued case to mappings between arbitrary metric spaces. Consider a function f: X \to Y, where (X, d_X) and (Y, d_Y) are metric spaces. The function f is bounded if there exists a finite constant M > 0 such that d_Y(f(x), f(x')) \leq M for all x, x' \in X.^[9] This condition ensures that the image of f has finite diameter in Y, generalizing the supremum bound on differences in the real-valued setting. Equivalently, f is bounded if its range f(X) is a bounded subset of Y, meaning f(X) is contained within a ball of finite radius in the metric d_Y.^[10] In normed vector spaces, where Y is equipped with a norm \|\cdot\|_Y, boundedness can be expressed as \sup_{x \in X} \|f(x)\|_Y < \infty, or more precisely, there exists M \geq 0 such that \|f(x)\|_Y \leq M for all x \in X. This formulation emphasizes that the values of f remain confined within a bounded region of the norm topology, independent of the domain's structure beyond being a set. For functions mapping to \mathbb{R}^n with the Euclidean norm, boundedness aligns with componentwise conditions. Specifically, if f = (f_1, \dots, f_n) where each f_i: X \to \mathbb{R} is bounded, then f is bounded in the Euclidean norm, since \|f(x)\|_2^2 = \sum_{i=1}^n f_i(x)^2 \leq n \max_i \sup_{x \in X} |f_i(x)|^2 < \infty. The converse holds as well, as each component satisfies |f_i(x)| \leq \|f(x)\|_2. While boundedness controls the extent of the function's values, it does not imply continuity or uniform continuity; for instance, the characteristic function of a nonempty proper subset of X is bounded but discontinuous at boundary points. Nonetheless, boundedness serves as a foundational condition in theorems concerning the compactness of function spaces, such as the Arzelà–Ascoli theorem, which characterizes relatively compact subsets of continuous functions on compact metric spaces as those that are uniformly equicontinuous and pointwise bounded.^[11]

Properties

Algebraic properties

Bounded functions form a vector space over the real or complex numbers under pointwise addition and scalar multiplication. Specifically, if f and g are bounded functions on a domain D, with |f(x)| \leq M_f and |g(x)| \leq M_g for all x \in D and some constants M_f, M_g \geq 0, then their sum h(x) = f(x) + g(x) is bounded, satisfying |h(x)| \leq M_f + M_g for all x \in D.^[12]^[13] Similarly, the scalar multiple k(x) = c f(x) for a constant c \in \mathbb{R} (or \mathbb{C}) is bounded with |k(x)| \leq |c| M_f.^[12]^[13] The set of bounded functions is also closed under pointwise multiplication. For the product p(x) = f(x) g(x), the inequality |p(x)| = |f(x)| |g(x)| \leq M_f M_g holds for all x \in D, establishing boundedness with bound M_p \leq M_f M_g.^[12] This follows directly from the properties of the absolute value, as |f| is bounded whenever f is, since ||f(x)|| = |f(x)| \leq M_f.^[12]^[13] Regarding compositions, if f: D \to \mathbb{R} is bounded and g: \mathbb{R} \to \mathbb{R} is continuous and thus bounded on bounded sets (such as the closed interval containing the image of f), then g \circ f is bounded on D.^[13] For restrictions to subsets, if a function is bounded on a subset S \subseteq D, it remains bounded when restricted to any subset of S, but boundedness on S does not necessarily extend to the full domain D unless the function is defined to remain controlled outside S; counterexamples exist where extensions beyond S render the function unbounded on D.^[12]

Analytic properties

In real analysis, a key connection between boundedness and continuity arises on compact domains. Specifically, if a function f: D \to \mathbb{R} is continuous on a compact set K \subseteq D, then f is bounded on K. This result, known as part of the Weierstrass theorem, follows from the fact that the image f(K) is also compact and hence bounded in \mathbb{R}./04:_Function_Limits_and_Continuity/4.08:_Continuity_on_Compact_Sets._Uniform_Continuity) Uniform continuity strengthens this relation on compact sets. A function that is uniformly continuous on a compact set K is necessarily continuous on K and thus bounded there, as uniform continuity implies continuity. However, the converse does not hold: boundedness does not imply uniform continuity, as demonstrated by bounded but discontinuous functions like the step function on [0,1], which is bounded yet fails uniform continuity due to a jump discontinuity./04:_Function_Limits_and_Continuity/4.08:_Continuity_on_Compact_Sets._Uniform_Continuity) The extreme value theorem provides a precise characterization of boundedness for continuous functions on closed bounded intervals. For a continuous function f: [a, b] \to \mathbb{R}, f attains its maximum and minimum values on [a, b], implying that f is bounded, with bounds given by \max\{|f(c)|, |f(d)|\} where c, d \in [a, b] are the points achieving the extrema. This attainment ensures the function's range is contained within [\min f, \max f], a finite interval./04:_Continuity/4.04:_The_Extreme_Value_Theorem) Boundedness also plays a central role in integrability criteria. A bounded function f: [a, b] \to \mathbb{R} is Riemann integrable if and only if it is continuous almost everywhere on [a, b], meaning the set of discontinuities has Lebesgue measure zero. This criterion highlights how boundedness, combined with limited discontinuities, ensures the upper and lower Riemann sums converge to the same value. Regarding limits, the existence of a finite limit implies local boundedness. If \lim_{x \to c} f(x) = L \in \mathbb{R} where c is an accumulation point of the domain, then there exists a neighborhood U of c such that f is bounded on U \cap D. This follows from the definition of the limit: for \epsilon = 1, a \delta > 0 ensures |f(x) - L| < 1 for $0 < |x - c| < \delta, x \in D, so |f(x)| < |L| + 1 nearby, with the value at c (if defined) also bounded.^[14] Finally, oscillation quantifies variation and ties directly to boundedness. The oscillation of f over an interval I, defined as \omega(f, I) = \sup_{x \in I} f(x) - \inf_{x \in I} f(x), is finite if and only if f is bounded on I. For unbounded functions, \omega(f, I) = \infty, reflecting infinite variation, whereas bounded functions exhibit controlled oscillation, bounded by twice the bound on |f|. Local oscillation at a point a, \omega(f, a) = \inf_{\delta > 0} \omega(f, (a - \delta, a + \delta) \cap D), is zero precisely when f is continuous at a.^[15]

Examples

Bounded functions

Constant functions provide the simplest examples of bounded functions. For any constant c \in \mathbb{R}, the function f(x) = c for all x in its domain satisfies |f(x)| = |c|, so it is bounded above by |c| and below by -|c|.^[16] Trigonometric functions such as sine and cosine are also bounded on the real line. The function f(x) = \sin x satisfies |\sin x| \leq 1 for all real x, as \sin x represents the y-coordinate on the unit circle, where the distance from the origin ensures the coordinate cannot exceed 1 in absolute value geometrically.^[17] Similarly, f(x) = \cos x satisfies |\cos x| \leq 1, corresponding to the x-coordinate on the same unit circle.^[18] Step functions illustrate boundedness in discontinuous cases. The Heaviside step function, defined as H(x) = 0 for x < 0 and H(x) = 1 for x \geq 0, is bounded by 1 on [0, \infty), where it takes the constant value 1.^[19] Certain rational functions are bounded even on unbounded domains. For instance, f(x) = \frac{1}{1 + x^2} on \mathbb{R} satisfies $0 < f(x) \leq 1, since x^2 \geq 0 implies $1 + x^2 \geq 1, so f(x) \leq 1 with equality at x = 0.^[20] Periodic continuous functions on \mathbb{R} are bounded if they are bounded on one period. Specifically, if f: \mathbb{R} \to \mathbb{R} is continuous and periodic with period p > 0, then restricting f to the compact interval [0, p] yields a continuous function that attains its maximum and minimum by the extreme value theorem; periodicity ensures these bounds hold globally.^[21] Sums of bounded functions are also bounded, as seen with \sin x + \cos x, which remains bounded despite the combination.

Unbounded functions

Unbounded functions are those that are not bounded, meaning there is no finite interval [M, N] such that M \leq f(x) \leq N for all x in the domain, often because the function diverges to \pm \infty at certain points or as the input approaches the boundary of the domain.^[22] This failure of boundedness typically arises from the function's growth behavior, which can vary in speed and direction. Polynomial functions provide classic examples of unboundedness on the real line. For instance, f(x) = x^n where n \geq 1 is unbounded on \mathbb{R} because as |x| \to \infty, |f(x)| \to \infty, with the degree n determining the rate of this polynomial growth.^[23] Similarly, exponential functions like f(x) = e^x on \mathbb{R} are unbounded above, as f(x) \to \infty when x \to \infty, while f(x) = -e^{-x} is unbounded below, since f(x) \to -\infty as x \to -\infty.^[24] Logarithmic functions, such as f(x) = \ln x on (0, \infty), are also unbounded, diverging to \infty as x \to \infty and to -\infty as x \to 0^+.^[23] Rational functions can exhibit unboundedness due to vertical asymptotes within or at the boundary of their domains. For example, f(x) = 1/x on (0, 1) is unbounded near x = 0, where f(x) \to \infty as x \to 0^+ because of the vertical asymptote at x = 0.^[22] These examples highlight distinct growth classifications: polynomial unboundedness grows relatively slowly compared to the rapid, suprapolynomial expansion of exponential functions, which eventually outpace any polynomial for large inputs.^[25]

Bounded sequences and series

In the context of sequences, a real-valued sequence \{a_n\}_{n=1}^\infty is bounded if there exists some M > 0 such that |a_n| \leq M for all n \in \mathbb{N}.^[26] This condition is equivalent to the range \{a_n : n \in \mathbb{N}\} forming a bounded subset of \mathbb{R}.^[26] Boundedness plays a key role in convergence properties; for instance, the Bolzano-Weierstrass theorem states that every bounded sequence in \mathbb{R} has a convergent subsequence.^[27] Cauchy sequences provide another connection to boundedness. Every Cauchy sequence \{a_n\} in \mathbb{R} is bounded. To verify this, fix \epsilon = 1; there exists N \in \mathbb{N} such that |a_m - a_n| < 1 whenever m, n \geq N. Setting m = N, it follows that |a_n| \leq |a_N| + 1 for all n \geq N. The finitely many terms a_1, \dots, a_{N-1} are bounded by some finite maximum, so the entire sequence is bounded.^[28] For series, boundedness refers to the sequence of partial sums s_n = \sum_{k=1}^n a_k. A series \sum a_k is said to have bounded partial sums if there exists M > 0 such that |s_n| \leq M for all n \in \mathbb{N}. Bounded partial sums imply that conditional convergence is possible without absolute convergence; for example, conditionally convergent series like the alternating harmonic series \sum (-1)^{k+1}/k have convergent (hence bounded) partial sums, while \sum |(-1)^{k+1}/k| diverges.^[29] However, bounded partial sums do not guarantee convergence of the series. In contrast, unbounded partial sums indicate divergence, as seen in the harmonic series \sum 1/k, whose partial sums H_n satisfy \ln n < H_n < 1 + \ln n for n > 1 and thus grow without bound.^[30]

Bounded operators in functional analysis

In functional analysis, a linear operator T: X \to Y between normed vector spaces X and Y is bounded if there exists a constant M \geq 0 such that \|T x\|_Y \leq M \|x\|_X for all x \in X.^[31] This condition ensures that T maps bounded sets in X to bounded sets in Y, providing a measure of the operator's "size" or stability.^[32] The operator norm \|T\| is defined as the infimum of all such M, or equivalently,

\|T\| = \sup \left\{ \frac{\|T x\|_Y}{\|x\|_X} \mid x \in X, x \neq 0 \right\} = \sup \left\{ \|T x\|_Y \mid x \in X, \|x\|_X \leq 1 \right\},

which quantifies the maximum stretch induced by T.^[33] Boundedness is equivalent to continuity of T at the origin (and hence everywhere, by linearity), as continuity at zero implies the existence of such an M.^[34] Examples of bounded operators include the identity operator I: X \to X, which satisfies \|I x\| = \|x\| and thus has norm \|I\| = 1.^[35] Multiplication operators on L^p spaces provide another class: for a measurable function b with essential supremum \|b\|_\infty < \infty, the operator M_b f = b f on L^p(\mu) (where $1 \leq p \leq \infty) is bounded with \|M_b\| = \|b\|_\infty.^[36] These operators are fundamental in studying function spaces and spectral theory. A key result concerning families of bounded operators is the uniform boundedness principle, also known as the Banach-Steinhaus theorem: if X is a Banach space and \{T_\alpha: X \to Y\} is a family of bounded linear operators to a normed space Y that is pointwise bounded (i.e., \sup_\alpha \|T_\alpha x\|_Y < \infty for each x \in X), then the family is uniformly bounded, meaning \sup_\alpha \|T_\alpha\| < \infty.^[37] This theorem prevents pathological behaviors in infinite-dimensional spaces and has applications in approximation theory and duality.^[38]

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