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Is there a function which is continuous and differentiable, but is not smooth function?

By smooth I mean having continuous derivative. For example, the derivative of $f(x)=x|x|/2$ is $f'(x)=|x|$ which is continuous. So I consider this function smooth.


marked as duplicate by Hagen von Eitzen, Git Gud, zarathustra, graydad, Etienne Jan 14 '15 at 20:08

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  • 3
    $\begingroup$ Does $x\mapsto |x|$ have an antiderivative? $\endgroup$ – Git Gud Jan 14 '15 at 19:11
  • $\begingroup$ I think yes, x|x|/2 $\endgroup$ – user96634 Jan 14 '15 at 19:14
  • $\begingroup$ @GitGud Isn't that antiderivative smooth? $\endgroup$ – user96634 Jan 14 '15 at 19:15
  • 1
    $\begingroup$ What does it mean for a function to be smooth? $\endgroup$ – Git Gud Jan 14 '15 at 19:18
  • $\begingroup$ To have continuous derivatives. The derivative of x|x|/2 is |x| which is continuous. So I think it's smooth $\endgroup$ – user96634 Jan 14 '15 at 19:21

One standard example is $f(0) = 0$, $f(x) = x^2*\sin(1/x)$. Then $f'(0) = 0$, but $f'$ is not continuous at $0$.


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