Let $f: \mathbb R \to \mathbb R$ be a continuous function, and there exists a $k>0$ such that for each $y \in \mathbb R$, there are at most $k$ distinct $x$ with $f(x)=y$. Prove that $f$ is differentiable a.e.
My approach is trying to show $f$ is absolutly continuous, but need a hint to make a start. Thanks.