Editing Smoothness (section)

==Differentiability classes==
'''Differentiability class''' is a classification of functions according to the properties of their [[derivative]]s. It is a measure of the highest order of derivative that exists and is continuous for a function.

Consider an [[open set]] <math>U</math> on the [[real line]] and a function <math>f</math> defined on <math>U</math> with real values. Let ''k'' be a non-negative [[integer]]. The function <math>f</math> is said to be of differentiability '''class ''<math>C^k</math>''''' if the derivatives <math>f',f'',\dots,f^{(k)}</math> exist and are [[continuous function|continuous]] on <math>U.</math> If <math>f</math> is <math>k</math>-differentiable on <math>U,</math> then it is at least in the class <math>C^{k-1}</math> since <math>f',f'',\dots,f^{(k-1)}</math> are continuous on <math>U.</math> The function <math>f</math> is said to be '''infinitely differentiable''', '''smooth''', or of '''class <math>C^\infty,</math>''' if it has derivatives of all orders on <math>U.</math> (So all these derivatives are continuous functions over <math>U.</math>)<ref name="def diff">{{cite book| last=Warner| first=Frank W.| author-link=Frank Wilson Warner| year=1983| title=Foundations of Differentiable Manifolds and Lie Groups| publisher=Springer| isbn=978-0-387-90894-6| page=5 [Definition 1.2]| url=https://books.google.com/books?id=t6PNrjnfhuIC&dq=%22f+is+differentiable+of+class+Ck%22&pg=PA5| access-date=2014-11-28| archive-date=2015-10-01| archive-url=https://web.archive.org/web/20151001012659/https://books.google.com/books?id=t6PNrjnfhuIC&pg=PA5&dq=%22f+is+differentiable+of+class+Ck%22| url-status=live}}</ref> The function <math>f</math> is said to be of '''class <math>C^\omega,</math>''' or ''[[analytic function|analytic]]'', if <math>f</math> is smooth (i.e., <math>f</math> is in the class <math>C^\infty</math>) and its [[Taylor series]] expansion around any point in its domain converges to the function in some [[Neighbourhood (mathematics)|neighborhood]] of the point.  There exist functions that are smooth but not analytic; <math>C^\omega</math> is thus strictly contained in <math>C^\infty.</math> [[Bump function]]s are examples of functions with this property.

To put it differently, the class <math>C^0</math> consists of all continuous functions. The class <math>C^1</math> consists of all [[differentiable function]]s whose derivative is continuous; such functions are called ''continuously differentiable''. Thus, a <math>C^1</math> function is exactly a function whose derivative exists and is of class <math>C^0.</math> In general, the classes <math>C^k</math> can be defined [[recursion|recursively]] by declaring <math>C^0</math> to be the set of all continuous functions, and declaring <math>C^k</math> for any positive integer <math>k</math> to be the set of all differentiable functions whose derivative is in <math>C^{k-1}.</math> In particular, <math>C^k</math> is contained in <math>C^{k-1}</math> for every <math>k>0,</math> and there are examples to show that this containment is strict (<math>C^k \subsetneq C^{k-1}</math>). The class <math>C^\infty</math> of infinitely differentiable functions, is the intersection of the classes <math>C^k</math> as <math>k</math> varies over the non-negative integers.

===Examples===
==== Example: continuous (''C''<sup>0</sup>) but not differentiable ====
[[Image:C0 function.svg|thumb|The ''C''<sup>0</sup> function {{nowrap|1={{mvar|f}}({{mvar|x}}) = {{mvar|x}}}} for {{nowrap|{{mvar|x}} ≥ 0}} and 0 otherwise.]]
[[File:X^2sin(x^-1).svg|thumb|The function {{nowrap|1={{mvar|g}}({{mvar|x}}) = {{mvar|x}}<sup>2</sup> sin(1/{{mvar|x}})}} for {{nowrap|{{mvar|x}} &gt; 0}}.]]
[[File:The function x^2*sin(1 over x).svg|thumb|upright=1.3|The function <math>f:\R\to\R</math> with <math>f(x)=x^2\sin\left(\tfrac 1x\right)</math> for <math>x\neq 0</math> and <math>f(0)=0</math> is differentiable. However, this function is not continuously differentiable.]]
[[File:Mollifier Illustration.svg|thumb|upright=1.2|A smooth function that is not analytic.]]
The function
<math display="block">f(x) = \begin{cases}x  & \mbox{if } x \geq 0, \\ 0 &\text{if } x < 0\end{cases}</math>
is continuous, but not differentiable at {{nowrap|1={{mvar|x}} = 0}}, so it is of class ''C''<sup>0</sup>, but not of class ''C''<sup>1</sup>.

==== Example: finitely-times differentiable (''C''<sup>{{mvar|k}}</sup>) ====
For each even integer {{mvar|k}}, the function
<math display="block">f(x)=|x|^{k+1}</math>
is continuous and {{mvar|k}} times differentiable at all {{mvar|x}}. At {{nowrap|1={{mvar|x}} = 0}}, however, <math>f</math> is not {{nowrap|({{mvar|k}} + 1)}} times differentiable, so <math>f</math> is of class ''C''<sup>{{mvar|k}}</sup>, but not of class ''C''<sup>{{mvar|j}}</sup> where {{nowrap|{{mvar|j}} > {{mvar|k}}}}.

==== Example: differentiable but not continuously differentiable (not ''C''<sup>1</sup>)====
The function
<math display="block">g(x) = \begin{cases}x^2\sin{\left(\tfrac{1}{x}\right)} & \text{if }x \neq 0, \\ 0 &\text{if }x = 0\end{cases}</math>
is differentiable, with derivative
<math display="block">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}</math>

Because <math>\cos(1/x)</math> oscillates as {{mvar|x}} → 0, <math>g'(x)</math> is not continuous at zero. Therefore, <math>g(x)</math> is differentiable but not of class ''C''<sup>1</sup>.

==== Example: differentiable but not Lipschitz continuous ====
The function
<math display="block">h(x) = \begin{cases}x^{4/3}\sin{\left(\tfrac{1}{x}\right)} & \text{if }x \neq 0, \\ 0 &\text{if }x = 0\end{cases}</math>
is differentiable but its derivative is unbounded on a [[compact set]]. Therefore, <math>h</math> is an example of a function  that is differentiable but not locally [[Lipschitz continuous]].

==== Example: analytic (''C''<sup>{{mvar|ω}}</sup>) ====
The [[exponential function]] <math>e^{x}</math> is [[Analytic function|analytic]], and hence falls into the class ''C''<sup>ω</sup> (where ω is the smallest [[transfinite ordinal]]). The [[trigonometric function]]s are also analytic wherever they are defined, because they are [[Trigonometric_functions#Euler's_formula_and_the_exponential_function | linear combinations of complex exponential functions]] <math>e^{ix}</math> and <math>e^{-ix}</math>.

==== Example: smooth (''C''<sup>{{mvar|∞}}</sup>) but not analytic (''C''<sup>{{mvar|ω}}</sup>) ====
The [[bump function]]
<math display="block">f(x) = \begin{cases}e^{-\frac{1}{1-x^2}} & \text{ if } |x| < 1, \\ 0 &\text{ otherwise }\end{cases}</math>
is smooth, so of class ''C''<sup>∞</sup>, but it is not analytic at {{nowrap|1={{mvar|x}} = ±1}}, and hence is not of class ''C''<sup>ω</sup>. The function {{mvar|f}} is an example of a smooth function with [[compact support]].

===Multivariate differentiability classes===
A function <math>f:U\subseteq\mathbb{R}^n\to\mathbb{R}</math> defined on an open set <math>U</math> of <math>\mathbb{R}^n</math> is said<ref>{{cite book|author=Henri Cartan|title=Cours de calcul différentiel|year=1977|publisher=Paris: Hermann|author-link=Henri Cartan}}</ref> to be of class <math>C^k</math> on <math>U</math>, for a positive integer <math>k</math>, if all [[partial derivatives]] 
<math display="block">\frac{\partial^\alpha f}{\partial x_1^{\alpha_1} \, \partial x_2^{\alpha_2}\,\cdots\,\partial x_n^{\alpha_n}}(y_1,y_2,\ldots,y_n)</math>
exist and are continuous, for every <math>\alpha_1,\alpha_2,\ldots,\alpha_n</math> non-negative integers, such that <math>\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\leq k</math>, and every <math>(y_1,y_2,\ldots,y_n)\in U</math>. Equivalently, <math>f</math> is of class <math>C^k</math> on <math>U</math> if the <math>k</math>-th order [[Fréchet derivative]] of <math>f</math> exists and is continuous at every point of <math>U</math>. The function <math>f</math> is said to be of class <math>C</math> or <math>C^0</math> if it is continuous on <math>U</math>. Functions of class <math>C^1</math> are also said to be ''continuously differentiable''.

A function <math>f:U\subset\mathbb{R}^n\to\mathbb{R}^m</math>, defined on an open set <math>U</math> of <math>\mathbb{R}^n</math>, is said to be of class <math>C^k</math> on <math>U</math>, for a positive integer <math>k</math>, if all of its components
<math display="block">f_i(x_1,x_2,\ldots,x_n)=(\pi_i\circ f)(x_1,x_2,\ldots,x_n)=\pi_i(f(x_1,x_2,\ldots,x_n)) \text{ for } i=1,2,3,\ldots,m</math>
are of class <math>C^k</math>, where <math>\pi_i</math> are the natural [[Projection (linear algebra)|projections]] <math>\pi_i:\mathbb{R}^m\to\mathbb{R}</math> defined by <math>\pi_i(x_1,x_2,\ldots,x_m)=x_i</math>. It is said to be of class <math>C</math> or <math>C^0</math> if it is continuous, or equivalently, if all components <math>f_i</math> are continuous, on <math>U</math>.

===The space of ''C''<sup>''k''</sup> functions===
Let <math>D</math> be an open subset of the real line. The set of all <math>C^k</math> real-valued functions defined on <math>D</math> is a [[Fréchet space|Fréchet vector space]], with the countable family of [[seminorm]]s
<math display="block">p_{K, m}=\sup_{x\in K}\left|f^{(m)}(x)\right|</math>
where <math>K</math> varies over an increasing sequence of [[compact set]]s whose [[union (set theory)|union]] is <math>D</math>, and <math>m=0,1,\dots,k</math>.

The set of <math>C^\infty</math> functions over <math>D</math> also forms a Fréchet space. One uses the same seminorms as above, except that <math>m</math> is allowed to range over all non-negative integer values.

The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of [[partial differential equation]]s, it can sometimes be more fruitful to work instead with the [[Sobolev space]]s.