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What is the meaning of a function $f : \mathbb{R}^N \to \mathbb{R}$ being $\mathcal{C^k}$ where $k \in \mathbb{N}$? I need an explanation for the case $N \geq 2$.

2 Answers 2

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All partial derivatives up to order $k$ exist and are continuous.

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    Is there any restriction as to what happens to partial derivatives of order higher than $k$ ?2011-04-15
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    Also wonder what happens to the case of $f : \mathbb{R^N} \to \mathbb{R^M}$ ?2011-04-15
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    @Rajesh, no restriction. For the general case, you impose the conditions on all coordinate functions of $f$.2011-04-15
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    by no restriction do you mean one or more of them may not exist at some point. Please clarify whether i am getting it correct.2011-04-15
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    @Rajesh, being $\mathcal C^k$ has *no* implication on being $\mathcal C^r$ for $r>k$. In other words, $\mathcal C^r \subset \mathcal C^k$ and the inclusion is strict.2011-04-15
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The function is $k$ times differentiable everywhere on $\mathbb{R}^n$, and $D^kf$ is continuous on $\mathbb{R}^n$.

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    But defining $D^k f$ when $N>1$ makes this answer harder to understand than the lhf answer...2011-04-15