# Differentiable continuous function whose derivative is not continuous [duplicate]

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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, EtienneJan 14 '15 at 20:08

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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

## 1 Answer

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