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Smoothness

In mathematics, smoothness primarily refers to the property of functions or mappings being infinitely differentiable, meaning that derivatives of all orders exist and are continuous on their domain, ensuring no abrupt changes or discontinuities in behavior.^[1] This concept, often denoted as C^\infty, distinguishes smooth functions from those that are merely finitely differentiable, such as C^k functions where only up to the k-th derivative is continuous.^[2] Examples include the exponential function e^x and trigonometric functions like \sin x, which possess derivatives of every order that remain bounded and continuous.^[3] Beyond real analysis, smoothness extends to geometric and algebraic contexts, where it describes structures free of irregularities or singularities. In differential geometry, a smooth manifold is a topological space that locally resembles Euclidean space through smooth coordinate charts, with transition functions between charts being infinitely differentiable diffeomorphisms; this allows for the consistent definition of tangent spaces and differential forms across the manifold.^[2] For instance, the sphere S^2 admits a smooth structure via stereographic projections, enabling the study of geodesics and curvatures without "kinks."^[2] In algebraic geometry, smoothness characterizes varieties or schemes that are locally like affine space, specifically when all local rings are regular (i.e., the dimension equals the minimal number of generators of the maximal ideal).^[4] A morphism of schemes is smooth if it is flat, of finite presentation, and has geometrically smooth fibers, implying that the target variety inherits a structure amenable to resolution of singularities and deformation theory.^[5] This notion ensures that smooth varieties behave well under operations like intersection and allow for the application of powerful tools such as the implicit function theorem in the algebraic setting.^[5] Smoothness also appears in optimization and numerical analysis, where a function is termed L-smooth if its gradient is Lipschitz continuous with constant L, bounding the rate of change and facilitating convergence guarantees for algorithms like gradient descent.^[6] Across these fields, the unifying theme is the absence of pathological features, promoting tractability in theoretical and computational studies.

Fundamental Definitions

Definition and Historical Context

In mathematical analysis, smoothness refers to the property of a function being infinitely differentiable. Specifically, for a function f: U \to \mathbb{R}^m where U \subseteq \mathbb{R}^n is an open set, f is smooth (or C^\infty) if all iterated Fréchet derivatives D^k f exist and are continuous on U for every order k = 1, 2, 3, \dots. The Fréchet derivative generalizes the classical derivative to multivariable settings, representing the best linear approximation at each point, with higher-order derivatives defined recursively on these linear maps. This definition extends the notion of continuity (as the C^0 case) to arbitrary finite or infinite orders of differentiability. The historical roots of smoothness trace back to the 17th and 18th centuries, when Gottfried Wilhelm Leibniz introduced higher-order differentials in his foundational work on calculus around 1675–1690, using them to describe rates of change beyond the first order. Leonhard Euler built on this in the mid-18th century, employing higher derivatives extensively in his analyses of series expansions and differential equations, such as in his Institutiones calculi integralis (1768–1770), where he explored repeated integration and differentiation intuitively without full rigor.^[7] These early contributions treated higher differentiability as a natural extension of basic calculus, often in the context of solving physical problems like trajectories and vibrations. The 19th century brought rigorous formalization, driven by the need to address foundational issues in analysis. Augustin-Louis Cauchy, in his 1821 Cours d'analyse de l'École Royale Polytechnique, provided the first precise definition of the derivative via limits and extended it systematically to higher orders, defining the k-th derivative as the limit of appropriate difference quotients and proving continuity of derivatives under suitable conditions. Karl Weierstrass complemented this in the 1860s through his lectures in Berlin, emphasizing the epsilon-delta formalism for limits, which ensured the consistency of differentiability classes and highlighted pathologies like nowhere-differentiable continuous functions. These developments established smoothness as a cornerstone of real analysis, distinguishing it from mere continuity. In the 20th century, smoothness gained deeper structure through functional analysis. Stefan Banach's 1932 Théorie des opérations linéaires introduced Banach spaces, paving the way for studying spaces of smooth functions, such as C^\infty(U), equipped with seminorms that make them complete Fréchet spaces. This framework formalized infinite-order differentiability in infinite-dimensional settings. Notably, while smoothness universally denotes C^\infty properties, in partial differential equations (PDEs), "regularity" specifically describes how elliptic or parabolic operators bootstrap solutions to higher smoothness levels from initial data, assuming only finite differentiability.^[8]

Differentiability in One Variable

A function f: I \to \mathbb{R}, where I is an open interval in \mathbb{R}, is differentiable at a point c \in I if the limit

f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}

exists and is finite.^[9] This definition captures the instantaneous rate of change of f at c, and the function is differentiable on I if it is differentiable at every point in I. For example, polynomial functions such as f(x) = x^2 are differentiable everywhere, with f'(x) = 2x.^[9] Higher-order derivatives are defined recursively: if f^{(n)} exists on an open interval, then f^{(n+1)}(x) = \frac{d}{dx} f^{(n)}(x) at points where the derivative exists.^[9] This iterative process allows for the study of successively finer approximations to the function's behavior. Taylor's theorem provides a framework for such approximations, stating that 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) in Lagrange form is

R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x - a)^{n+1}

for some \xi between a and x.^[10] This quantifies the error in the polynomial approximation, essential for understanding local behavior near a. The chain rule facilitates differentiation of composite functions: if f is differentiable at g(x) and g is differentiable at x, then (f \circ g)'(x) = f'(g(x)) \cdot g'(x).^[11] Open intervals as domains ensure that limits can be approached from both sides, supporting these operations at interior points. Smoothness builds on these concepts through repeated differentiability.

Smoothness Classes

Finite-Order Differentiability (C^k)

The space C^k(\mathbb{R}) consists of all real-valued functions f: \mathbb{R} \to \mathbb{R} that are k-times differentiable, with each derivative f^{(j)} for $0 \leq j \leq k existing and continuous everywhere on \mathbb{R}. This class captures functions with a finite level of smoothness, where the k-th derivative is continuous but higher derivatives may not exist or be continuous.^[12] To endow C^k(\mathbb{R}) with a topological structure, it is equipped with the norm

\|f\|_{C^k} = \max_{0 \leq j \leq k} \sup_{x \in \mathbb{R}} |f^{(j)}(x)|,

which requires all derivatives up to order k to be bounded (hence the space is a proper subspace of all k-times continuously differentiable functions). This norm induces a metric, and under this metric, C^k(\mathbb{R}) is a Banach space: every Cauchy sequence converges to an element in the space. Completeness follows from the fact that uniform limits preserve continuity and differentiability up to order k; specifically, if \{f_n\} is Cauchy in the C^k-norm, then each \{f_n^{(j)}\} for j \leq k converges uniformly to a continuous function g_j, and by standard calculus results, g_j = ( \lim f_n )^{(j)}. Closed linear subspaces of C^k(\mathbb{R}) inherit the Banach space properties.^[12]^[13] The spaces satisfy strict inclusion relations: C^{k+1}(\mathbb{R}) \subset C^k(\mathbb{R}) for each k \geq 0, with the inclusion map being continuous (i.e., \|f\|_{C^k} \leq \|f\|_{C^{k+1}}). On compact subsets K \subset \mathbb{R}, the restrictions of polynomials are dense in the restricted C^k(K) under the C^k-norm. This density follows from the Stone-Weierstrass theorem, which guarantees polynomial density in C^0(K), extended iteratively: a C^k function can be approximated by integrating approximations of its derivatives, yielding polynomial approximations that converge in the C^k-norm.^[12]^[14]

Infinite Differentiability (C^∞)

A function f: \Omega \to \mathbb{R}, where \Omega \subseteq \mathbb{R} is open, is said to be infinitely differentiable if it belongs to the class C^k(\Omega) for every nonnegative integer k, meaning all derivatives up to order k exist and are continuous on \Omega. Equivalently, the space C^\infty(\Omega) is the intersection \bigcap_{k=0}^\infty C^k(\Omega).^[15] When \Omega is a compact interval, C^\infty(\Omega) is endowed with a Fréchet space topology generated by the countable family of seminorms

\|f\|_m = \sup_{x \in \Omega} \max_{0 \leq k \leq m} |f^{(k)}(x)|

for m = 0, 1, 2, \dots. This topology is complete and metrizable, ensuring uniform convergence of functions and all their derivatives up to any fixed order.^[16] The space C^\infty(\mathbb{R}) is endowed with the Fréchet topology generated by the countable family of seminorms p_n(f) = \max_{0 \leq k \leq n} \sup_{x \in [-n,n]} |f^{(k)}(x)| for n = 1, 2, \dots . This topology is complete and metrizable, ensuring uniform convergence of functions and all their derivatives up to any fixed order on every compact subset of \mathbb{R}. In the theory of distributions, the subspace \mathcal{D}(\mathbb{R}) = C_c^\infty(\mathbb{R}) of compactly supported infinitely differentiable functions is equipped with a similar LF-space topology as the inductive limit of C^\infty([-n,n]), serving as the standard space of test functions whose continuous linear dual is the space of distributions.^[17] To prepare for multivariable extensions, the topology on C^\infty(\Omega) for \Omega \subseteq \mathbb{R}^n open is the Fréchet topology induced by the family of seminorms

\|f\|_{K,m} = \sup_{x \in K} \sup_{|\alpha| \leq m} |D^\alpha f(x)|

over all compact subsets K \subset \Omega and orders m \in \mathbb{N}, where \alpha is a multi-index. This topology is complete and metrizable when defined via a countable basis of seminorms.

Analytic Smoothness (C^ω)

Analytic smoothness, denoted as the class C^\omega, refers to functions that are infinitely differentiable and moreover locally equal to their Taylor series expansions. A real-valued function f: U \to \mathbb{R}, where U \subset \mathbb{R} is open, is said to be analytic at a point a \in U if there exists some r > 0 such that the Taylor series of f around a,

\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n,

converges to f(x) for all x in the interval (a - r, a + r) \cap U. This requires the power series to have a positive radius of convergence, ensuring local representation by a convergent power series. The function f belongs to C^\omega(U) if it is analytic at every point in U.^[18] This local power series representation distinguishes C^\omega functions from the broader class of infinitely differentiable functions. Analyticity implies infinite differentiability, as term-by-term differentiation of the convergent power series yields the higher derivatives within the radius of convergence. However, the converse does not hold: C^\omega is a strict subclass of C^\infty, meaning there exist functions that are infinitely differentiable everywhere but fail to equal their Taylor series in any neighborhood of some points.^[19] A key characterization of analyticity involves bounds on the growth of derivatives. For instance, if the derivatives at a satisfy |f^{(n)}(a)| \leq M \frac{n!}{r^n} for all n, some constants M > 0 and r > 0, then the Taylor series converges absolutely in |x - a| < r, and by standard theorems on power series, it equals f(x) there, confirming analyticity. This estimate mirrors Cauchy's bounds derived from complex integral representations, which can be adapted to real functions via analytic continuation or majorant series. Classic examples of C^\omega functions include polynomials, which are analytic everywhere with finite Taylor series (higher derivatives vanish), and entire functions like the exponential f(x) = e^x. The Taylor series of e^x around any point a is \sum_{n=0}^\infty \frac{(x - a)^n}{n!}, which converges to e^x for all real x, demonstrating global analyticity on \mathbb{R}. Similarly, \sin x and \cos x are analytic on \mathbb{R}, with their series converging everywhere. These functions highlight how C^\omega captures rigid structures governed by power series, underpinning applications in approximation theory and differential equations.^[19]

Examples and Illustrations

Continuous Functions Without Higher Differentiability

A fundamental example of a function that is continuous everywhere but fails to be differentiable at a specific point is the absolute value function f(x) = |x|. This function belongs to the class C^0(\mathbb{R}), meaning it is continuous on the real line, but it is not differentiable at x = 0 because the left-hand derivative is -1 while the right-hand derivative is +1, so the derivative does not exist there.^[20] Despite this lack of differentiability at the origin, |x| remains uniformly continuous on \mathbb{R} since it satisfies the Lipschitz condition with constant 1, implying ||x| - |y|| \leq |x - y| for all x, y \in \mathbb{R}.^[20] Far more pathological are functions that are continuous everywhere but differentiable nowhere, such as the Weierstrass function introduced by Karl Weierstrass in 1872. This function is defined by the infinite series

w(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x),

where $0 < a < 1, b is an odd positive integer, and ab > 1 + \frac{3\pi}{2}.^[21] Weierstrass's construction ensures uniform convergence of the series on \mathbb{R}, guaranteeing continuity everywhere, yet the rapid oscillations induced by the parameters prevent the existence of a derivative at any point due to the failure of the difference quotient to converge.^[21] The function's fractal-like graph, with self-similar wiggles at every scale, exemplifies its nowhere-differentiable nature, challenging the 18th-century intuition that continuous functions should be smooth.^[22] Such examples highlight the boundary between mere continuity and differentiability, revealing that continuity alone does not imply even local smoothness. While the absolute value function's kink is intuitive and isolated, the Weierstrass function's total absence of tangents underscores the existence of highly irregular yet continuous behaviors in real analysis. Both functions are uniformly continuous on compact intervals, preserving some regularity, but their non-differentiability illustrates how pathological constructions can evade higher smoothness without violating continuity.^[21]^[20]

Functions with Limited Smoothness Orders

Functions with limited smoothness orders refer to those that are exactly C^k for some finite k > 0, meaning they have continuous derivatives up to order k, but the (k+1)-th derivative either does not exist or is not continuous at some points. These examples illustrate how smoothness can break down at specific orders, providing counterexamples to the idea that higher differentiability automatically follows from lower orders. Such functions are crucial in real analysis for demonstrating the sharpness of differentiability classes.^[3] A representative example of a function that is C^1 but not C^2 is f(x) = \frac{1}{2} x |x|, defined for all real x. This function is continuously differentiable with f'(x) = |x|, which is continuous everywhere. However, the second derivative f''(x) = \operatorname{sign}(x) for x \neq 0 does not exist at x = 0, as the left and right derivatives differ. This shows a discontinuity in the existence of higher derivatives at a point. Another classic construction involves oscillation to achieve limited smoothness. Consider f(x) = x^3 \sin(1/x) for x \neq 0 and f(0) = 0. This function is C^1, with f'(x) = 3x^2 \sin(1/x) - x \cos(1/x) for x \neq 0 and f'(0) = 0, and f' is continuous at 0 since both terms vanish in the limit. The second derivative exists for x \neq 0 but the limit defining f''(0) oscillates and does not exist due to the \cos(1/x) term, confirming it is not twice differentiable at 0. This example highlights how rapid oscillations can prevent higher-order differentiability while preserving lower-order continuity. Regarding Lipschitz continuity, differentiability does not imply the derivative is bounded, leading to functions that are differentiable but not Lipschitz continuous on bounded intervals. An example is g(x) = x^2 \sin(1/x^2) for x \neq 0 and g(0) = 0. This is differentiable everywhere, with g'(x) = 2x \sin(1/x^2) - (2/x) \cos(1/x^2) for x \neq 0 and g'(0) = 0, but |g'(x)| can be as large as approximately $2/|x| near 0, making the derivative unbounded on any interval containing 0. Consequently, g fails to be Lipschitz continuous near 0, as the mean value theorem would require bounded slopes for such continuity.^[23] For higher finite orders, one can construct functions that are C^k but not C^{k+1} by repeated integration of nowhere-differentiable continuous functions like the Weierstrass function w(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x) with $0 < a < 1 and ab > 1 + 3\pi/2, which is continuous everywhere but differentiable nowhere. The k-fold integral f(x) = \int_0^x \int_0^{t_{k-1}} \cdots \int_0^{t_1} w(t) \, dt yields a function in C^k but not C^{k+1}, as each integration increases the smoothness order by 1, but the final derivative is w, which lacks differentiability. This method generalizes to arbitrary finite k, emphasizing the role of pathological base functions in limiting smoothness.

Non-Analytic Smooth Functions

A canonical example of a non-analytic smooth function is the flat function at the origin, given by

f(x) = \begin{cases} \exp\left( -\frac{1}{x^2} \right) & x > 0, \\ 0 & x \leq 0. \end{cases}

This function belongs to the class C^\infty(\mathbb{R}), as it is infinitely differentiable everywhere, including at x=0, where all derivatives vanish: f^{(n)}(0) = 0 for every n \geq 0. The Taylor series of f centered at 0 is thus the zero polynomial, which converges pointwise to the zero function on \mathbb{R}, but fails to equal f(x) for any x > 0 where f(x) > 0. Consequently, f is not analytic at 0, despite its smoothness there; the radius of convergence of the Taylor series is infinite, yet it does not represent f in any neighborhood of 0. This example, first identified by Cauchy in 1823 as a C^\infty function with peculiar behavior at a point, underscores the strict inclusion C^\omega \subsetneq C^\infty.^[24] The existence of such functions reveals a fundamental distinction from analytic functions, where the Taylor series always converges to the function in some neighborhood of the expansion point. The Denjoy-Carleman theorem characterizes Denjoy-Carleman classes C^M (subclasses of C^\infty with derivative growth bounded by |f^{(n)}| \leq C^{n+1} M_n) as quasi-analytic—meaning functions agreeing to all orders at a point coincide nearby—if \sum_{n=1}^\infty (M_n / M_{n+1})^{1/n} = \infty. The analytic class C^\omega (with M_n \sim n!) satisfies this (sum diverges), hence quasi-analytic. In contrast, the full C^\infty class admits non-quasi-analytic behavior, as exemplified by the flat function: its zero Taylor series at 0 matches the zero function, yet f differs nearby. Borel's theorem further illuminates this gap by establishing the surjectivity of the Taylor expansion map from C^\infty(\mathbb{R}) onto the space of formal power series: for any sequence (a_n), there exists a smooth function whose nth derivative at 0 is n! a_n. The flat function demonstrates the map's non-injectivity, as distinct smooth functions can share the same Taylor series. These results highlight how smoothness permits greater flexibility in derivative behavior than analyticity demands, with quasi-analyticity serving as a bridge between the two. A symmetric extension, f(x) = \exp(-1/x^2) for x \neq 0 and 0 at 0, is also smooth and flat at 0, and such constructions yield bump functions supported on compact sets, useful in advanced applications like manifolds.

Multivariable and Parametric Smoothness

Partial Derivatives and Multivariable Classes

In multivariable calculus, differentiability of a function f: \mathbb{R}^n \to \mathbb{R}^m at a point x \in \mathbb{R}^n is defined using the Fréchet derivative, which is a linear map Df(x): \mathbb{R}^n \to \mathbb{R}^m such that the limit \lim_{h \to 0} \frac{\|f(x + h) - f(x) - Df(x)(h)\|}{\|h\|} = 0 holds, providing a best linear approximation to f near x.^[25] This derivative is represented by the Jacobian matrix, whose entries are the partial derivatives \frac{\partial f_i}{\partial x_j}(x), generalizing the single-variable derivative to higher dimensions.^[26] The smoothness classes C^k extend to multivariable functions by requiring that all partial derivatives up to order k exist and are continuous on an open set U \subset \mathbb{R}^n.^[27] To compactly denote these higher-order partials, multi-index notation is used: a multi-index \alpha = (\alpha_1, \dots, \alpha_n) is a tuple of non-negative integers with |\alpha| = \sum_{i=1}^n \alpha_i, and the partial derivative D^\alpha f = \frac{\partial^{|\alpha|} f}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}.^[28] A function f belongs to C^k(U) if D^\alpha f is continuous for all \alpha with |\alpha| \leq k; when n=1, this reduces to the one-variable case.^[29] The higher-order chain rule for compositions of multivariable functions, such as f(g(x)) where g: \mathbb{R}^n \to \mathbb{R}^p and f: \mathbb{R}^p \to \mathbb{R}^m, is given by the multivariate Faà di Bruno formula, which expresses the k-th derivative as a sum over partitions involving products of derivatives of f and g.^[30] This formula generalizes the univariate chain rule and accounts for all possible ways partial derivatives combine under composition.^[30] The multivariable Taylor theorem approximates a C^k function f: \mathbb{R}^n \to \mathbb{R} near a point a by

f(x) = \sum_{|\alpha| \leq k} \frac{D^\alpha f(a)}{\alpha!} (x - a)^\alpha + R_k(x),

where \alpha! = \prod_{i=1}^n \alpha_i! is the multi-index factorial and R_k(x) is the remainder term, often bounded by a form involving the (k+1)-th derivatives.^[31] This expansion relies on the continuity of partials up to order k and provides local polynomial approximations in multiple variables.^[32]

Parametric and Geometric Continuity

In the context of parametrized curves and surfaces, parametric continuity of order k, denoted C^k, requires that the parametrization \gamma: I \to \mathbb{R}^n (where I is an interval and n \geq 2) is k-times continuously differentiable, meaning the derivatives \gamma^{(0)}, \gamma^{(1)}, \dots, \gamma^{(k)} are all continuous functions on I. For piecewise parametrizations, such as those used in spline representations, C^k continuity at junction points demands that the position and all derivatives up to order k match exactly between adjacent segments, ensuring both geometric and parametric smoothness without abrupt changes in speed or acceleration.^[33] This strict condition is valuable in applications like computer-aided design (CAD) for generating curves where precise control over the parametrization's velocity is needed, as in uniform B-spline curves that inherently achieve C^{k-1} continuity for degree-k polynomials. Geometric continuity of order k, denoted G^k, relaxes the parametric requirement by allowing a local reparametrization—typically an affine transformation of the parameter—that renders the curve C^k smooth. Introduced to address limitations in parametric continuity, where geometrically smooth shapes might fail C^k due to incompatible parametrizations, G^k focuses on the intrinsic geometry rather than the specific speed of traversal.^[33] For instance, G^0 coincides with C^0, requiring only positional continuity for a continuous curve without cusps. At order 1, G^1 ensures tangent vector directions align at junctions (no kinks), but magnitudes may differ, permitting flexible spline designs like beta-splines in CAD systems where shape control parameters adjust curvature without violating smoothness. Higher orders, such as G^2, extend this to continuous curvature by aligning osculating planes and curvatures up to scalar multiples, facilitating fairer surfaces in modeling applications.^[33] The distinction between C^k and G^k is particularly pronounced in spline-based modeling, where C^k enforces rigid derivative matching that can constrain design freedom, whereas G^k enables more intuitive geometric constructions, such as blending curves with varying arc-length parametrizations. In CAD software, G^1 continuity is often sufficient for visual smoothness in automotive or aerospace design, avoiding the computational overhead of full C^1 while maintaining manufacturable surfaces; for example, NURBS patches commonly achieve G^1 across edges via knot multiplicity adjustments.^[33] Three equivalent characterizations of G^n—via affine reparametrizations, linear dependence of derivative vectors, and matching Taylor expansions up to order n—provide practical tests for implementation in geometric modeling algorithms.

Advanced Structures and Properties

Smooth Functions on Manifolds

A smooth structure on a topological manifold M is defined by a maximal atlas \mathcal{A} = \{(U_\alpha, \phi_\alpha)\}_{\alpha \in I}, where each U_\alpha is an open subset of M, each \phi_\alpha: U_\alpha \to \mathbb{R}^n is a homeomorphism onto an open set in \mathbb{R}^n, and the transition maps \phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta) are C^\infty diffeomorphisms whenever U_\alpha \cap U_\beta \neq \emptyset.^[34] This maximal atlas ensures that the notion of smoothness is independent of the choice of charts within the equivalence class, allowing consistent differentiation across the manifold.^[2] The smooth structure equips M with a differentiable framework that generalizes the C^\infty category from Euclidean spaces to abstract spaces. A function f: M \to \mathbb{R} is smooth if, for every point p \in M, there exists a chart (U, \phi) containing p such that the composition f \circ \phi^{-1}: \phi(U) \to \mathbb{R} is a C^\infty function on the open subset \phi(U) \subseteq \mathbb{R}^n. This local criterion ensures that smoothness is well-defined globally on M, as the C^\infty property of transition maps guarantees compatibility between overlapping charts.^[34] Consequently, the space of smooth functions on M, often denoted C^\infty(M), forms a ring under pointwise addition and multiplication, serving as the foundation for differential geometry on manifolds.^[2] For maps between smooth manifolds F: M \to N, where M has dimension n and N has dimension m, F is smooth if for every p \in M, there exist charts (U, \phi) around p on M and (V, \psi) around F(p) on N such that the composition \psi \circ F \circ \phi^{-1}: \phi(U) \to \psi(V) is C^\infty.^[2] This definition extends the local smoothness condition to inter-manifold mappings, enabling the study of derivatives via tangent spaces. The tangent space T_p M at p \in M is the vector space of derivations on C^\infty(M) at p, or equivalently, in local coordinates, the space \mathbb{R}^n with the standard basis from the chart. The differential dF_p: T_p M \to T_{F(p)} N is the unique linear map such that for any smooth function g: N \to \mathbb{R}, (g \circ F)_*(p) = dF_p \circ ( )_p, where ( )_* denotes the derivation; in coordinates, it is the Jacobian matrix of \psi \circ F \circ \phi^{-1} at \phi(p).^[2] Partitions of unity can be used to extend local smooth functions to global ones on paracompact manifolds.^[35]

Partitions of Unity and Bump Functions

Bump functions are infinitely differentiable functions with compact support, meaning they are zero outside a bounded closed set and non-zero within an open subset of that set. These functions are essential tools in analysis and geometry for constructing smooth objects with controlled support. A canonical example in \mathbb{R}^n is given by the function \psi: \mathbb{R}^n \to \mathbb{R} defined as

\psi(x) = \begin{cases} \exp\left( -\frac{1}{1 - \|x\|^2} \right) & \text{if } \|x\| < 1, \\ 0 & \text{if } \|x\| \geq 1, \end{cases}

which can be normalized by dividing by its maximum value to ensure $0 \leq \psi(x) \leq 1 and \psi(0) = 1. This construction ensures all derivatives vanish at the boundary of the unit ball, preserving smoothness.^[36] Partitions of unity extend this idea to decompose the constant function 1 into a sum of smooth functions each supported in prescribed open sets. Formally, given an open cover \{U_i\}_{i \in I} of a topological space X, a partition of unity subordinate to \{U_i\} is a locally finite collection of smooth functions \{\rho_i\}_{i \in I}: X \to [0,1] such that \sum_{i \in I} \rho_i(x) = 1 for all x \in X and \operatorname{supp}(\rho_i) \subset U_i for each i. The local finiteness means that every point in X has a neighborhood intersecting only finitely many supports. This structure allows gluing local data into global smooth objects.^[37] The existence of smooth partitions of unity holds on paracompact smooth manifolds. Specifically, every paracompact Hausdorff smooth manifold admits a smooth partition of unity subordinate to any open cover. The proof relies on constructing bump functions in local charts and using mollifiers—smooth approximations to the Dirac delta via convolution with compactly supported functions—or approximate identities to smoothen step functions while preserving support properties. This theorem, building on earlier topological results, enables key constructions like extending smooth functions from closed subsets to the entire manifold.^[38]^[39]

Relation to Analyticity and Topology

In the theory of smooth functions, the concept of quasi-analyticity addresses the boundary between infinite differentiability and analyticity. A Denjoy-Carleman class C\{M_n\}, defined by a sequence of positive numbers M_n controlling the growth of higher derivatives via |f^{(n)}(x)| \leq M_n for functions f in the class, is quasi-analytic if the only function in the class that vanishes to infinite order at a point (i.e., all derivatives zero there) is the zero function everywhere in the connected component. The Denjoy-Carleman theorem characterizes such classes precisely: the class is quasi-analytic if and only if \sum_{n=1}^\infty M_n^{-1/n} = \infty. For the standard analytic functions, where M_n = n!, the series diverges like the harmonic series, ensuring quasi-analyticity and thus uniqueness from derivatives, akin to Taylor series convergence. In contrast, faster-growing sequences like M_n = (n!)^2 yield non-quasi-analytic classes, allowing non-zero smooth functions that are flat (infinitely differentiable but all derivatives zero) at a point. Smoothness also interacts deeply with topology through embedding theorems and the existence of exotic structures. The Whitney embedding theorem asserts that any smooth n-dimensional manifold admits a smooth embedding into \mathbb{R}^{2n}, preserving the smooth structure compatibly with the topological embedding into Euclidean space. This compatibility highlights how smoothness refines topological manifolds by providing a differential structure that allows local Euclidean charts with smooth transition maps. However, smoothness is not uniquely determined by topology: John Milnor's 1956 discovery revealed exotic smooth structures on the 7-sphere, yielding multiple non-diffeomorphic smooth manifolds homeomorphic to the standard S^7. Extending this, \mathbb{R}^4 admits uncountably many pairwise non-diffeomorphic smooth structures, all homeomorphic to the standard \mathbb{R}^4, demonstrating that topological and smooth categories diverge significantly in dimension 4.^[40]^[41] In complex analysis, the relation between smoothness and analyticity contrasts sharply with the real case. A holomorphic function, defined as complex differentiable in a domain, is automatically infinitely differentiable (smooth) as a real map and moreover real-analytic, with its Taylor series converging to the function locally.^[42] This rigidity arises from the Cauchy-Riemann equations and Cauchy's integral formula, which enforce strong control on derivatives. In the real setting, however, C^\infty functions need not be analytic, permitting non-analytic examples despite infinite differentiability; quasi-analytic classes, via the Denjoy-Carleman theorem, delineate precisely when real smoothness forces analytic-like uniqueness, partially bridging this gap between real flexibility and complex rigidity.

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[PDF] 14. Smoothness Definition - UCSD Math
Smoothness. Definition 14.1. A variety is smooth (aka non-singular) if all of its local rings are regular local rings. Theorem 14.2.Missing: mathematics | Show results with:mathematics
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Section 29.34 (01V4): Smooth morphisms—The Stacks project
We say that f is smooth if it is smooth at every point of X. A morphism of affine schemes f : X \to S is called standard smooth if there exists a standard ...
[6]
[PDF] MS&E 213 / CS 269O : Chapter 2 Smooth Functions
Oct 17, 2020 · Primarily this section is meant to introduce smoothness, a natural assumption we will use on objective functions, and gradient descent, an ...
[7]
[PDF] Chapter 4: Elliptic PDEs - UC Davis Math
Roughly speaking, solutions of elliptic PDEs are as smooth as the data allows. For boundary value problems, it is convenient to consider the regularity of the.
[8]
Derivative -- from Wolfram MathWorld
The derivative of a function represents an infinitesimal change in the function with respect to one of its variables.
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Taylor's Theorem -- from Wolfram MathWorld
Taylor's theorem states that any function satisfying certain conditions may be represented by a Taylor series, Taylor's theorem (without the remainder term) ...
[10]
Chain Rule -- from Wolfram MathWorld
If g(x) is differentiable at the point x and f(x) is differentiable at the point g(x), then f degreesg is differentiable at x. Furthermore, let y=f(g(x)) ...
[11]
[PDF] Banach Spaces - UC Davis Math
More generally, the space C(K) of continuous functions on a compact metric space K equipped with the sup-norm is a Banach space.
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[PDF] Banach spaces Ck[a, b]
Nov 13, 2016 · This function space can be presented as. C∞[a, b] = \ k≥0. Ck[a, b] and we reasonably require that whatever topology C∞[a, b] should have ...
[13]
Why are polynomials dense in $C^k(\bar{B}) - Math Stack Exchange
Dec 6, 2017 · asks if the set of all polynomials are dense in Ck(ˉB) ... You can see this argument in Rudin's PMA in proving Weierstrass in one dimension.Are the algebraic functions dense in the space of analytic functions?Real Polynomials on Compact sets of Complex numbersMore results from math.stackexchange.com
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About the definition of class $C^\infty - Math Stack Exchange
Feb 4, 2020 · But since Ck⊇Ck+1 for all k, we can just as easily define C∞=∞⋂k=0CkorC∞=∞⋂k=2CkorC∞=∞⋂k=47Ck. When taking an intersection over a decreasing ...C∞ dense in Sobolev spaces - Math Stack ExchangeCan differentiability classes be extended to negative values?More results from math.stackexchange.com
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Good metric on $C^k(0,1)$ and $C^\infty(0,1) - Math Stack Exchange
Jan 30, 2013 · The space C∞(0,1) is defined to be the space of all real-valued functions f on (0,1) with the property that Dkf∈C(0,1) for all k∈N0. Choosing ...Missing: LF- | Show results with:LF-
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[PDF] Examples of function spaces 1. Non-Banach limits C k(R), C (R) of ...
Feb 11, 2017 · Many familiar and useful spaces of continuous or differentiable functions, such as Ck[a, b], have natural metric structures, and are complete.
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[PDF] Distributions: Topology and Sequential Compactness.
An LF-space is also called the strict inductive limit of Fréchet spaces. If ... Then the sequence pn will tend to φ in the topology on C∞(K). Since ...
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[PDF] Multiplications and Convolutions in L. Schwartz'Spaces of Test ...
A seminorm on a function space merely measures the derivatives of a function up to a certain order, but for a C∞–function f and a distribution T, the distribu-.
[19]
Real Analysis: 8.4. Taylor Series - MathCS.org
Definition 8.4.3: Taylor Series ... A function f that can be expressed as a power series with non-empty radius of convergence is called a real analytic function.
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[PDF] Chapter 1: Preliminaries - UC Davis Math
A real-analytic function is C∞, since its Taylor series can be differentiated term-by-term, but a C∞ function need not be real-analytic. For example, see ...
[21]
[PDF] Differentiability
at x = 0 does not exist. Thus the absolute value function is not differentiable at x = 0. We can also see this by looking at the graph of |x|: at x = 0, the ...<|control11|><|separator|>
[22]
[PDF] Weierstrass's non-differentiable function
Dec 12, 2014 · In 1872, Karl Weierstrass shocked the mathe- matical world by giving the first published example of a continuous function that is nowhere.
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The Jagged, Monstrous Function That Broke Calculus
Jan 23, 2025 · In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.”
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A differentiable real function with unbounded derivative around zero
Dec 4, 2016 · On the interval (−1,1), g(x) is bounded by 2. However, for ak=1√kπ with k∈N we have h(ak)=2√kπ(−1)k which is unbounded while limk→∞ak=0.Missing: example | Show results with:example
[25]
The Origin and Early Development of Non-Analytic Infinitely ... - jstor
The origin of infinitely differentiable functions (referred to henceforth as C°° functions) not analytic at some point lies quite naturally in the answer to the.
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Fréchet Derivative -- from Wolfram MathWorld
The Fréchet derivative is sometimes known as the strong derivative (Ostaszewski 2012) and can be seen as a generalization of the gradient to arbitrary vector ...Missing: multivariable textbook
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[PDF] Multivariable Calculus Lectures Richard J. Brown
We will not do it in class. Definition 5.3. A function f ∶ X ⊂ R n → R is of class Ck, k ∈ N if it has continuous partial derivative up to and including order k ...
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2.5: Higher Order Derivatives
Multi-index notation for higher derivatives. If f is a Ck function of n variables, then a kth-order partial derivative of f can always be written in the form ...Missing: smoothness | Show results with:smoothness
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[PDF] Multi-index notation - Applied Mathematics Consulting
Sep 18, 2008 · Multi-index notation makes multi-variable generalizations of familiar one- variable theorems easier to remember. We will give two examples: ...
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[PDF] 1 Recap 2 Review of multivariable calculus - IISc Math
Using the C2 Clairaut, we can prove the Ck Clairaut for any k. Theorem 1 (Taylor). Let U ⊂ Rn be an open set and f : U → R be a Ck function on U. Let.
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the-multivariate-faa-di-bruno-formula-and-multivariate-taylor ...
The Faa di Bruno formula is for higher-order derivatives of composite functions, and the multivariate version is a generalization of the univariate case.
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[PDF] Math 396. Higher derivatives and Taylor's formula via multilinear maps
There is also a more traditional version of the multivariable Taylor formula given with loads of mixed partials and factorials, and we will show that this ...
[33]
[PDF] Proof of Taylor's Theorem Comments on notation: Suppose α = (α1 ...
Proof of Taylor's Theorem. Comments on notation: Suppose α = (α1,α2,...,αn) is a multi-index. The length of α is. |α| = α1 + ··· + αn, and α! is defined to be ...Missing: reference | Show results with:reference
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[PDF] Geometric Continuity of Parametric Curves - UC Berkeley EECS
In this paper, we extend the notions of geometric continuity to obtain G" continuity, for an arbitrary integer n ≥ 0. We show that for each level of ...
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[PDF] 1.4 Smooth Manifolds Defined
Let U CR and VCRm be open sets. A map f: UV is called smooth iff it is infinitely differentiable, i.e. iff all its partial derivatives.
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[PDF] Chapter 2: Smooth maps - Arizona Math
Sep 21, 2010 · In this chapter we introduce smooth maps between manifolds, and some impor- tant concepts. Definition 1. A function f : M ! Rk is a smooth ...Missing: mathematics | Show results with:mathematics
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bump function in nLab
Sep 12, 2024 · References. 1. Idea. A bump function is a smooth function with compact support, especially one that is not zero on a space that is not compact.Missing: seminal paper
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partition of unity in nLab
Jun 13, 2025 · Existence on smooth manifolds. Paracompact smooth manifolds admit locally finite smooth partitions of unity subordinate to any open cover ...Missing: seminal | Show results with:seminal
[39]
[PDF] Partitions of unity, supports of distributions 1. Paracompactness 2 ...
Dec 18, 2022 · . [3.1] Theorem: Partitions of unity exist. That is, on smooth, countably-based manifolds, there exists a. (smooth, locally finite) partition ...Missing: seminal reference
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[PDF] lecture 3: smooth functions
Example (Bump function). A bump function (sometimes also called a test function) is a compactly supported smooth function, which is usually supposed to be non- ...
[41]
[PDF] On manifolds homeomorphic to the 7-sphere
ON MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE. BY JOHN MILNOR¹. (Received June 14, 1956). The object of this note will be to show that the 7-sphere possesses ...
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[PDF] Milnor's construction of exotic 7-spheres
(4) R4 has uncountably many exotic smooth structures. No other Euclidean spaces have any exotic smooth structures. (5) We can explicitly write down an ...