Proposition 8

For a finite real function $f$ to be convex (resp. strictly convex) on an open interval $I \subset \mathbf{R}$ it is necessary and sufficient that it be continuous on $I$, have a derivative at every point of the complement $B$ relative to $I$ of a countable subset of this interval, and that the derivative be increasing (resp. strictly increasing) on $B$.

The condition is necessary, from prop. 6 and its corollary 2 (I, p. 27); let us show that it is sufficient. Suppose, therefore, that $f'$ is increasing on $B$, and that $f$ is not convex; there then exist (I, p. 27, prop. 5) three points $a, b, c$ of $I$, such that $a < c < b$, and $\frac{f(c) - f(a)}{c - a} > \frac{f(b) - f(c)}{b - c}$; but from the mean value theorem (I, p. 14, th. 1) one has

One thus has $\sup_{\lambda \in B,\ a < \lambda < c} f'(\lambda) > \inf_{\lambda \in B,\ c < \lambda < b} f'(\lambda)$, contrary to the hypothesis that $f'$ is increasing on $B$.

If now we assume that $f'$ is strictly increasing on $B$, then $f$ is convex and cannot be equal to an affine linear function on any open interval contained in $I$, for then $f'$ would be constant on this interval, contrary to the hypothesis.