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Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is differentiable? If yes, please provide some examples.

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    $\begingroup$ Your second example has a continuous derivative, actually. But the first is an example of an everywhere differentiable function whose derivative is discontinuous. $\endgroup$ May 4, 2015 at 15:20
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    $\begingroup$ What is your definition of differentiable? $\endgroup$ May 4, 2015 at 15:22
  • $\begingroup$ It's possible using some piece-wise definitions, at least. See here $\endgroup$ May 4, 2015 at 15:22
  • $\begingroup$ As an aside, see Volterra's function. $\endgroup$
    – Lucian
    May 4, 2015 at 16:19

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The functions you mentioned are in fact differentiable, so you can use them as examples.

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