Differentiable continuous function whose derivative is not continuous

Question

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.

Answer 1

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