Editing Differentiable function (section)

==Differentiability classes==
[[File:Approximation of cos with linear functions without numbers.svg|300px|thumb|Differentiable functions can be locally approximated by linear functions.]]
[[File:The function x^2*sin(1 over x).svg|thumb|300px|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.]]
{{main|Smoothness}}
A function <math display="inline">f</math> is said to be {{em|{{visible anchor|continuously differentiable|Continuous differentiability}}}} if the derivative <math display="inline">f^{\prime}(x)</math> exists and is itself a continuous function. Although the derivative of a differentiable function never has a [[jump discontinuity]], it is possible for the derivative to have an [[Classification of discontinuities#Essential discontinuity|essential discontinuity]]. For example, the function
<math display="block">f(x) \;=\; \begin{cases} x^2 \sin(1/x) & \text{ if }x \neq 0 \\ 0 & \text{ if } x = 0\end{cases}</math>
is differentiable at 0, since
<math display="block">f'(0) = \lim_{\varepsilon \to 0} \left(\frac{\varepsilon^2\sin(1/\varepsilon)-0}{\varepsilon}\right) = 0</math>
exists. However, for <math>x \neq 0,</math> [[differentiation rules]] imply
<math display="block">f'(x) = 2x\sin(1/x) - \cos(1/x)\;,</math>
which has no limit as <math>x \to 0.</math> Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless, [[Darboux's theorem (analysis)|Darboux's theorem]] implies that the derivative of any function satisfies the conclusion of the [[intermediate value theorem]].

Similarly to how [[continuous function]]s are said to be of {{em|class <math>C^0,</math>}} continuously differentiable functions are sometimes said to be of {{em|class <math>C^1</math>}}. A function is of {{em|class <math>C^2</math>}} if the first and [[second derivative]] of the function both exist and are continuous. More generally, a function is said to be of {{em|class <math>C^k</math>}} if the first <math>k</math> derivatives <math display="inline">f^{\prime}(x), f^{\prime\prime}(x), \ldots, f^{(k)}(x)</math> all exist and are continuous. If derivatives <math>f^{(n)}</math> exist for all positive integers <math display="inline">n,</math> the function is [[Smooth function|smooth]] or equivalently, of {{em|class <math>C^{\infty}.</math>}}
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