PL-OM 1977 Problem 4

A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$ , where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$ . Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$ . Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$ .