Exercise 23, § 4

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