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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

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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 $f'(x)$.

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    $\begingroup$ let $f(x)=|x|$ then $f$ is continuous at $0$, but not differentiable. Ok $\endgroup$ – Muniain Oct 8 '13 at 15:57
  • $\begingroup$ @Muniain There you go, you've answered your question :) $\endgroup$ – Neal Oct 10 '13 at 2:49

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