Exercise 13, § 6

Let $f$ be a continuous finite real-valued function on a compact interval $[a, b]$ of $\mathbf{R}$, and suppose that $f$ is not strictly monotone. Show that there exists a point $x_0 \in ]a, b[$ such that, for each $\varepsilon > 0$, there are two points $y, z$ in $[a, b]$ such that $x_0 - \varepsilon < y < x_0 < z < x_0 + \varepsilon$ and $f(y) = f(z)$.