Define f(x) = x² sin(1/x) for x ≠ 0 and f(0) = 0. Which statement is correct at x = 0?

Since $|x^2\sin(1/x)|\le x^2$, $\lim_{x\to0}f(x)=0=f(0)$, so $f$ is continuous. Also, $f'(0)=\lim_{h\to0}\frac{h^2\sin(1/h)}{h}=\lim_{h\to0}h\sin(1/h)=0$. For $x\ne0$, $f'(x)=2x\sin(1/x)-\cos(1/x)$, which has no limit as $x\to0$; hence $f'$ is not continuous at 0.