Editing Smoothness (section)

===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]].