Corollary 2

If $f$ is convex (resp. strictly convex) on I then $f'_d$ and $f'_g$ are increasing (resp. strictly increasing) on the interior of I; the set of points in I at which $f$ is not differentiable is countable, and $f'_d$ and $f'_g$ are continuous at every point where $f$ is differentiable.

The first part follows immediately from (8) (resp. (9)) and the inequality

On the other hand, let E be the set of interior points $x$ of I where $f$ is not differentiable (that is $f'_g(x) < f'_d(x)$). For each $x \in E$ let $J_x$ be the open interval $]f'_g(x), f'_d(x)[$; it follows from (8) that if $x$ and $y$ are two points of E such that $x < y$, then $u < v$ for all $u \in J_x$ and all $v \in J_y$; in other words, as $x$ runs through E the open nonempty intervals $J_x$ are pairwise disjoint; the set of such intervals is thus countable, and hence so is E. Finally, $f'_d$ (resp. $f'_g$) being increasing, it has a right limit and a left limit at every interior point $x$ of I; prop. 6 of I, p. 18 now shows that the right limit of $f'_d$ (resp. $f'_g$) at the point $x$ is equal to $f'_d(x)$, and its left limit is $f'_g(x)$, from which we have the last part of the corollary.

Let $f$ be a convex function on I, $a$ an interior point of I, and D a line passing through the point $M_a$, with equation $y - f(a) = \alpha(x - a)$. It follows from the inequalities (8) that if $f'_g(a) \leq \alpha \leq f'_d(a)$ then every point of the graph G lies above D, and, if $f$ is strictly convex, $M_a$ is the only point common to D and G; one says that D is a support line to G at the point $M_a$. Conversely, if G lies above D, one has $f(x) - f(a) \geq \alpha(x - a)$ for every $x \in I$, from which $\frac{f(x) - f(a)}{x - a} \geq \alpha$ for $x > a$, and $\frac{f(x) - f(a)}{x - a} \leq \alpha$ for $x < a$; on letting $x$ tend to $a$ in these inequalities it follows that $f'_g(a) \leq \alpha \leq f'_d(a)$.

In particular, if $f$ is differentiable at the point $a$ there is only one supporting line to G at the point $M_a$, the tangent to G at $M_a$.