Exercise 23, § 4

Let $f$ be a continuous real function on a compact interval $I \subset \mathbf{R}$; among the functions $g \leq f$ which are convex on $I$ there exists one, $g_0$, larger than all the others. Let $F \subset I$ be the set of $x \in I$ where $g_0(x) = f(x)$; show that $F$ is not empty and that on each of the open intervals contiguous to $F$ the function $g_0$ is equal to an affine linear function (argue by contradiction).