Sometimes I've encountered functions [itex]f:\mathbb{R}^n\to\mathbb{R}^m[/itex] being called N-times differentiable. What does it mean, precisely?(adsbygoogle = window.adsbygoogle || []).push({});

I know that for a function to be differentiable, if is not enough that the partial derivatives [itex]\partial_i f_j[/itex] exist, but instead the derivative matrix [itex]Df[/itex] must exist and satisfy the equation

[tex]

f(x+u)=f(x) + (Df)u + |u|\epsilon(u)

[/tex]

which is more than only existence of the partial derivatives.

Does the N-times differentiability mean something else than only all partial derivatives [itex]\partial_1^{k_1} \partial_2^{k_2} \cdots \partial_n^{k_n} f_j[/itex], where [itex]k_1+k_2+\cdots+k_n=N[/itex], existing?

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# Multiple times differentiable

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