Exercise 1, § 6

Let $f$ be a continuous mapping of an open interval $I \subset \mathbf{R}$ into $\mathbf{R}$. Show that if $f(I)$ is open and if, for each $y \in \mathbf{R}$, the set $f^{-1}(y)$ has at most two distinct points, then $f$ is monotone.