Smoothness
In mathematical analysis, the smoothness describes the number of times a function can be differentiated without producing discontinuities. The smoothness, or differentiability class, is an integer such that a function has all derivatives up to order , and such that all of these derivatives are continuous. One says that such a function has class . For example, the absolute value function has class , because it is continuous, but not differentiable. Generally, the term smooth function refers to a -function, that is a function having derivatives of all orders. However, it may also mean "sufficiently differentiable" for the problem under consideration.
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Differentiability class is a classification of functions according to the highest order of derivative that exists and is continuous for a function. Consider an open set U {\displaystyle U} on the real line and a function f {\displaystyle f} defined on U {\displaystyle U} with real values. Let k be a non-negative integer. The function f {\displaystyle f} is said to be of differentiability class C k {\displaystyle C^{k}} if the derivatives f ′ , f ″ , … , f ( k ) {\displaystyle f',f'',\dots ,f^{(k)}} exist and are continuous on U . {\displaystyle U.} If f {\displaystyle f} is of class C k {\displaystyle C^{k}} on U {\displaystyle U} and k > 0 {\displaystyle k>0} , then it is also of class C k − 1 {\displaystyle C^{k-1}} . The function f {\displaystyle f} is said to be infinitely differentiable, smooth, or of class C ∞ , {\displaystyle C^{\infty },} if it is of class C k {\displaystyle C^{k}} for every non-negative integer k {\displaystyle k} . The function f {\displaystyle f} is said to be of class C ω , {\displaystyle C^{\omega },} or analytic, if f {\displaystyle f} is smooth and its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. There exist functions that are smooth but not analytic; C ω {\displaystyle C^{\omega }} is thus strictly contained in C ∞ . {\displaystyle C^{\infty }.} Bump functions are examples of functions with this property.
Examples
The function f ( x ) = { x if x ≥ 0 , 0 if x < 0 {\displaystyle f(x)={\begin{cases}x&{\mbox{if }}x\geq 0,\\0&{\text{if }}x<0\end{cases}}} is continuous, but not differentiable at x = 0, so it is of class C0, but not of class C1. For each even non-negative integer k, the function f ( x ) = | x | k + 1 {\displaystyle f(x)=|x|^{k+1}} is continuous and of class C k {\displaystyle C^{k}} . At x = 0, however, f {\displaystyle f} is not of class C k + 1 {\displaystyle C^{k+1}} , so f {\displaystyle f} is of class Ck, but not of class Cj where j > k. The function g ( x ) = { x 2 sin ( 1 x ) if x ≠ 0 , 0 if x = 0 {\displaystyle g(x)={\begin{cases}x^{2}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}} is differentiable, with derivative g ′ ( x ) = { − cos ( 1 x ) + 2 x sin ( 1 x ) if x ≠ 0 , 0 if x = 0. {\displaystyle g'(x)={\begin{cases}-{\mathord {\cos \left({\tfrac {1}{x}}\right)}}+2x\sin \left({\tfrac {1}{x}}\right)&{\text{if }}x\neq 0,\\0&{\text{if }}x=0.\end{cases}}}
Multivariate differentiability classes
A function f : U ⊆ R n → R {\displaystyle f:U\subseteq \mathbb {R} ^{n}\to \mathbb {R} } defined on an open set U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} is said to be of class C k {\displaystyle C^{k}} on U {\displaystyle U} , for a positive integer k {\displaystyle k} , if all partial derivatives D α f = ∂ | α | f ∂ x 1 α 1 ∂ x 2 α 2 ⋯ ∂ x n α n {\displaystyle D^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\,\partial x_{2}^{\alpha _{2}}\,\cdots \,\partial x_{n}^{\alpha _{n}}}}} exist and are continuous for every multi-index α = ( α 1 , α 2 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})} of non-negative integers with | α | = α 1 + α 2 + ⋯ + α n ≤ k {\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k} . Equivalently, in finite dimensions, f {\displaystyle f} is of class C k {\displaystyle C^{k}} on U {\displaystyle U} if it is k {\displaystyle k} times continuously Fréchet differentiable on U {\displaystyle U} . The function f {\displaystyle f} is said to be of class C {\displaystyle C} or C 0 {\displaystyle C^{0}} if it is continuous on U {\displaystyle U} . Functions of class C 1 {\displaystyle C^{1}} are also said to be continuously differentiable.
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Open domains
Let D {\displaystyle D} be an open subset of R n {\displaystyle \mathbb {R} ^{n}} . The set of all real-valued C k {\displaystyle C^{k}} functions on D {\displaystyle D} is denoted C k ( D ) {\displaystyle C^{k}(D)} . With the compact-open C k {\displaystyle C^{k}} topology, C k ( D ) {\displaystyle C^{k}(D)} is a Fréchet space. One way to describe this topology is by the family of seminorms p K , α ( f ) = sup x ∈ K | D α f ( x ) | , {\displaystyle p_{K,\alpha }(f)=\sup _{x\in K}|D^{\alpha }f(x)|,} where K {\displaystyle K} ranges over compact subsets of D {\displaystyle D} and α {\displaystyle \alpha } ranges over multi-indices with | α | ≤ k {\displaystyle |\alpha |\leq k} .
Compact domains
If U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} is bounded and open, then C k ( U ¯ ) {\displaystyle C^{k}({\overline {U}})} denotes the space of functions on U {\displaystyle U} whose partial derivatives of order at most k {\displaystyle k} extend continuously to the compact set U ¯ {\displaystyle {\overline {U}}} . It is a Banach space with the norm ‖ f ‖ C k ( U ¯ ) = max | α | ≤ k sup x ∈ U ¯ | D α f ( x ) | . {\displaystyle \|f\|_{C^{k}({\overline {U}})}=\max _{|\alpha |\leq k}\sup _{x\in {\overline {U}}}|D^{\alpha }f(x)|.} Equivalently, one may use the sum of these suprema over | α | ≤ k {\displaystyle |\alpha |\leq k} ; the resulting norm is equivalent.
Density
The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of partial differential equations, it can sometimes be more fruitful to work instead with Sobolev spaces. Smooth compactly supported functions are dense in many function spaces used in analysis, such as L p {\displaystyle L^{p}} spaces and Sobolev spaces under suitable hypotheses. These correspond to putting topologies on the smooth functions that are weaker than those of uniform convergence (like the L p {\displaystyle L^{p}} norm). This makes smooth functions useful as test functions and as approximations to less regular functions.
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The differentiability classes C k {\displaystyle C^{k}} are closed under the usual algebraic operations. If f {\displaystyle f} and g {\displaystyle g} are real-valued functions of class C k {\displaystyle C^{k}} on the same domain, then f + g {\displaystyle f+g} , f g {\displaystyle fg} , and any scalar multiple of f {\displaystyle f} are also of class C k {\displaystyle C^{k}} . If g {\displaystyle g} is nowhere zero, then the quotient f / g {\displaystyle f/g} is of class C k {\displaystyle C^{k}} . These facts follow from the sum, product, and quotient rules for derivatives. Moreover, the space C k ( U ) {\displaystyle C^{k}(U)} is a real vector space and, under pointwise multiplication, a commutative algebra. In particular, C ∞ ( M ) {\displaystyle C^{\infty }(M)} , the algebra of smooth real-valued functions on a smooth manifold M {\displaystyle M} , plays a central role in differential geometry: many geometric objects on M {\displaystyle M} can be described in terms of their action on smooth functions.
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For 0 < α ≤ 1 {\displaystyle 0<\alpha \leq 1} , the Hölder spaces C k , α ( U ) {\displaystyle C^{k,\alpha }(U)} on an open set U {\displaystyle U} in R n {\displaystyle \mathbb {R} ^{n}} are functions that are C k {\displaystyle C^{k}} on U {\displaystyle U} and whose k {\displaystyle k} -th partials satisfy a Hölder condition on U {\displaystyle U} : | ∂ k f ( x ) − ∂ k f ( y ) | ≤ C ‖ x − y ‖ α . {\displaystyle |\partial ^{k}f(x)-\partial ^{k}f(y)|\leq C\|x-y\|^{\alpha }.} This condition is stronger than ordinary continuity. When α = 1 {\displaystyle \alpha =1} , it implies the Lipschitz continuity of the k-th derivative, which is weaker than their differentiability. Thus, for 0 < α < 1 {\displaystyle 0<\alpha <1} , and on a non-empty open domain U {\displaystyle U} , C k ( U ) ⊊ C k , α ( U ) ⊊ C k , 1 ( U ) ⊊ C k + 1 ( U ) . {\displaystyle C^{k}(U)\subsetneq C^{k,\alpha }(U)\subsetneq C^{k,1}(U)\subsetneq C^{k+1}(U).}
Relation to analyticity
While all analytic functions are smooth on the set on which they are analytic, examples such as bump functions (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series; another example is the Fabius function. Although it might seem that such functions are the exception rather than the rule, analytic functions form a small subclass of smooth functions; for example, with suitable topologies on spaces of smooth functions, analytic functions form a meagre subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else.
Smoothness and the Fourier transform
Under suitable hypotheses, higher differentiability of a function is related to faster decay of its Laplace transform or Fourier transform. For example, integration by parts gives decay estimates for Fourier transforms of functions whose derivatives satisfy appropriate integrability or boundary conditions. These relationships are related to results such as the Paley–Wiener theorem. Conversely, decay of the Fourier transform can imply differentiability or continuity properties of the original function. This is often formulated using Sobolev spaces: Fourier-transform decay gives Sobolev regularity, and the Sobolev embedding theorem gives conditions under which Sobolev regularity implies classical C k {\displaystyle C^{k}} smoothness.
Test functions and distributions
Smooth compactly supported functions, usually denoted C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} , are called test functions. They are used to define distributions and weak derivatives.
Smooth partitions of unity
Smooth functions with suitably controlled support, especially smooth functions with compact support, are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [a,b] and such that f ( x ) > 0 for a < x < b . {\displaystyle f(x)>0\quad {\text{ for }}\quad a<x<b.\,} Given a locally finite collection of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals ( − ∞ , c ] {\displaystyle (-\infty ,c]} and [ d , + ∞ ) {\displaystyle [d,+\infty )} to cover the whole line, such that the sum of the functions is always 1.
Smooth functions on and between manifolds
Given a smooth manifold M {\displaystyle M} , of dimension m , {\displaystyle m,} and an atlas U = { ( U α , ϕ α ) } α , {\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },} a map f : M → R {\displaystyle f:M\to \mathbb {R} } is smooth on M {\displaystyle M} if, for every p ∈ M {\displaystyle p\in M} , there is a chart ( U , ϕ ) ∈ U , {\displaystyle (U,\phi )\in {\mathfrak {U}},} with p ∈ U , {\displaystyle p\in U,} such that f ∘ ϕ − 1 : ϕ ( U ) → R {\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} } is a smooth function from the open subset ϕ ( U ) {\displaystyle \phi (U)} of R m {\displaystyle \mathbb {R} ^{m}} to R {\displaystyle \mathbb {R} } . Similarly, f {\displaystyle f} is of class C k {\displaystyle C^{k}} if these coordinate representations are of class C k {\displaystyle C^{k}} . Smoothness can be checked with respect to any chart of the atlas that contains p , {\displaystyle p,} since the smoothness requirements on the transition functions between charts ensure that if f {\displaystyle f} is smooth near p {\displaystyle p} in one chart it will be smooth near p {\displaystyle p} in any other chart.
Smooth functions between subsets of manifolds
There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If f : X → Y {\displaystyle f:X\to Y} is a function whose domain and codomain are subsets of manifolds X ⊆ M {\displaystyle X\subseteq M} and Y ⊆ N {\displaystyle Y\subseteq N} , respectively, then f {\displaystyle f} is said to be smooth if for all x ∈ X {\displaystyle x\in X} there is an open set U ⊆ M {\displaystyle U\subseteq M} with x ∈ U {\displaystyle x\in U} and a smooth function F : U → N {\displaystyle F:U\to N} such that F ( p ) = f ( p ) {\displaystyle F(p)=f(p)} for all p ∈ U ∩ X . {\displaystyle p\in U\cap X.}
