Solutions

We have that

$$\begin{align*} f'(0)&=\lim_{h\to0}\frac{f(h)-f(0)}{h} \\ &=\lim_{h\to0}\frac{hg(h)-0g(0)}{h} \\ &=\lim_{h\to0}g(h) \\ &=g(0). \end{align*}$$

Since $g$ is continuous at $0$, hence $f$ is differentiable at $0$.