Editing Differentiable function (section)

==Differentiability in higher dimensions==

A [[function of several real variables]] {{math|'''f''': '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>}} is said to be differentiable at a point {{math|'''x'''<sub>0</sub>}} if [[there exists]] a [[linear map]] {{math|'''J''': '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>}} such that
:<math>\lim_{\mathbf{h}\to \mathbf{0}} \frac{\|\mathbf{f}(\mathbf{x_0}+\mathbf{h}) - \mathbf{f}(\mathbf{x_0}) - \mathbf{J}\mathbf{(h)}\|_{\mathbf{R}^{n}}}{\| \mathbf{h} \|_{\mathbf{R}^{m}}} = 0.</math>

If a function is differentiable at {{math|'''x'''<sub>0</sub>}}, then all of the [[partial derivative]]s exist at {{math|'''x'''<sub>0</sub>}}, and the linear map {{math|'''J'''}} is given by the [[Jacobian matrix]], an ''n'' × ''m'' matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the [[fundamental increment lemma]] found in single-variable calculus.

If all the partial derivatives of a function exist in a [[Neighbourhood (mathematics)|neighborhood]] of a point {{math|'''x'''<sub>0</sub>}} and are continuous at the point {{math|'''x'''<sub>0</sub>}}, then the function is differentiable at that point {{math|'''x'''<sub>0</sub>}}.

However, the existence of the partial derivatives (or even of all the [[directional derivative]]s) does not guarantee that a function is differentiable at a point. For example, the function {{math|''f'': '''R'''<sup>2</sup> → '''R'''}} defined by

:<math display="block">f(x,y) = \begin{cases}x & \text{if }y \ne x^2 \\ 0 & \text{if }y = x^2\end{cases}</math>

is not differentiable at {{math|(0, 0)}}, but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function

:<math>f(x,y) = \begin{cases}y^3/(x^2+y^2) & \text{if }(x,y) \ne (0,0) \\ 0 & \text{if }(x,y) = (0,0)\end{cases}</math>

is not differentiable at {{math|(0, 0)}}, but again all of the partial derivatives and directional derivatives exist.

{{See also|Multivariable calculus|Smoothness#Multivariate differentiability classes}}