A counterexample of a differentiable function on an interval whose derivative is discontinuous

Question

Let $f(x)$ be a real-valued differentiable function on an interval $I$ of $\mathbb R$, then is $f'(x)$ necessary to be continuous?

I don't think so and I'm trying to construct a counterexample. $\sin\frac{1}{x}$ may be the key element but $x\sin\frac{1}{x}$ failed me.

Answer 1

Let $f(x)=x^{2}\sin(1/x)$ for $x\ne 0$, $f(0)=0$, then $f'(0)=0$, $f'(x)=2x\sin(1/x)-\cos(1/x^{2})$, for $x\ne 0$. You may try to investigate $\lim_{x\rightarrow 0}f'(x)$ to see what is going on.