Remark

If $f$ is a strictly convex function on an open interval I then $f'_d$ is strictly increasing on I, so there are three possible cases, according to prop. 2 of I, p. 13: 1. $f$ is strictly decreasing on I; 2. $f$ is strictly increasing on I; 3. there is an $a \in I$ such that $f$ is strictly decreasing for $x \leq a$, and is strictly increasing for $x \geq a$.

When $f$ is convex on I, but not strictly convex, $f$ can be constant on an interval contained in I; let $J = ]a, b[$ be the largest open interval on which $f$ is constant (that is to say, the interior of the interval where $f'_d(x) = 0$); then $f$ is strictly decreasing on the interval formed by the points $x \in I$ such that $x \leq a$ (if it exists), strictly increasing on the interval formed by the points $x \in I$ such that $x \geq b$ (if it exists).

In all cases one sees that $f$ possesses a right limit at the left-hand endpoint of $I$ (in $\overline{\mathbf{R}}$), and a left limit at the right-hand endpoint; these limits may be finite or infinite (cf. I, p. 46, exerc. 5, 6 and 7). By abuse of language one sometimes says that the continuous function (with values in $\overline{\mathbf{R}}$), equal to $f$ on the interior of $I$, and extended by continuity to the endpoints of $I$, is convex on $\overline{I}$.