Exercise 12, § 4

Let $f$ be a numerical function continuous in an open interval $I \subset \mathbf{R}$; suppose that, for every pair $(a, b)$ of points of $I$ such that $a < b$, the graph of $f$ is either wholly above or wholly below the segment $M_aM_b$ in the interval $[a, b]$. Prove that $f$ is convex in all of $I$ or concave in all of $I$ (if, in $]a, b[$, there exists a point $c$ such that $M_c$ is strictly above the segment $M_aM_b$ prove that for every $x \in I$ such that $x > a$, the graph of $f$ is above the segment $M_aM_x$ in the interval $[a, x]$).