For $p \in[1,\infty]$,let $f_p:\mathbb{R}^m\to\mathbb{R},x\mapsto|x|_p$. Where is $f_p$ differentiable?

To find those points,I first considered at which points that all partial derivatives are continuous, then those points are definitely differentiable points. But this process may not find all differentiable points of $f_p$, since there exist points at which partial derivative all exist but may not all continuous. So at such case, how to decide whether $f_p$ is differentiable at it?

  • $\begingroup$ There are different answers depending on whether $p<1$ or $p>1$, and whether $p=\infty$ or not. $\endgroup$ – uniquesolution Apr 15 at 12:12
  • $\begingroup$ I have solved this question. $f_p$ continuously differentiable $\iff$ all its partial derivatives exists and continuous. partial derivative doesn't exist $\Longrightarrow$ $\partial f_p$ doesn't exist. $\endgroup$ – Tao X 5 hours ago

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