Solutions

We have

$$\lim_{k\to 0^-}\frac{h(a+k)-h(a)}{k}=\lim_{k\to 0^-}\frac{f(a+k)-f(a)}{k}$$

and

$$\lim_{k\to 0^+}\frac{h(a+k)-h(a)}{k}=\lim_{k\to 0^+}\frac{g(a+k)-g(a)}{k}.$$

Since both limits are equal, then $h$ is differentiable at $a$.