continuous implies frechet differentiable?

Question

I knew if $f$ is Frechet differentiable at $x$ then $f$ is continuous at $x$. But reverse, i.e. If $f$ is continuous at $x$ then $f$ is Frechet differentiable at $x$ true or false?. I think it is wrong, but I can't give a counterexample. Can anyone give me a counterexample?. Thanks

Neal · Answer 1 · 2013-10-10T02:48:51.380

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The space $\mathbb{R}$ with the usual absolute value is a normed vector space, and the Frechet derivative coincides with the ordinary derivative.[1] Now can you come up with continuous functions that are not (Frechet-)differentiable?

[1] If $f:\mathbb{R}\to\mathbb{R}$, then its Frechet derivative at $x$ is the bounded linear operator defined as multiplication by $f'(x)$.

edited Oct 10 '13 at 02:48

answered Oct 07 '13 at 04:43

Neal

32,659

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let $f(x)=|x|$ then $f$ is continuous at $0$, but not differentiable. Ok – Muniain Oct 08 '13 at 15:57
@Muniain There you go, you've answered your question :) – Neal Oct 10 '13 at 02:49

continuous implies frechet differentiable?

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