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Do functions exist, which are differentiable in a point, but not in a neighborhood of this point?

Is $e^{\frac{1}{W(x)-2}}$, where W is the Weierstrass function, maybe an example of a such function?

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  • $\begingroup$ I think $x^2$ times the Dirichlet function D(x) has this property at x=0 $\endgroup$ – Maxim Umansky Jul 16 '13 at 4:32
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    $\begingroup$ I finally remembered this old math joke your name alludes to: what's yellow, normed, and complete? $\endgroup$ – Julien Jul 16 '13 at 7:22
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If we multiply the Weierstrass function by $x^2$, we get a function which is continuous everywhere, and differentiable at $0$ but nowhere else.

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  • $\begingroup$ Thanks for all responses. I set this answer as my accepted answer because I was looking for a continuous function. Sorry for not being clear. $\endgroup$ – Bananach Jul 16 '13 at 13:43
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The function $$f(x)=\chi_{\Bbb Q}(x)\cdot x^2$$ is differentiable (and continuous) only at $x=0$.

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  • $\begingroup$ Simplified your example. Please revert if you disagree. $\endgroup$ – Did Jul 16 '13 at 7:15
  • $\begingroup$ @Did I don't. In fact, what I wrote before was unnecessary. $\endgroup$ – Pedro Tamaroff Jul 16 '13 at 7:18
  • $\begingroup$ The solution in the book jumps from [$(v_n)$ converges to $v$] to [$f(v)=v$]. This step only holds if $f$ is continuous (at $v$). Hence, either the assumption that $f$ is continuous is in the book, or it was forgotten. $\endgroup$ – Did Jul 16 '13 at 8:35
  • $\begingroup$ Oh, and +1 for the solution here. $\endgroup$ – Did Jul 16 '13 at 8:36

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