Let $f(x)=x^2\sin(1/x)$ for $x≠ 0$ and $f(0)=0$ for $x=0$.
Is f' continuous at $0$?
My attempt: f'(x)=2x\sin(1/x)-\cos(1/x). Since when $x$ goes to $0$, the limit of $\cos(1/x)$ does not exist, it is not continuous. But I'm not sure since we did define $f(0)=0$...