Proposition 6

Let $f$ be a finite real function convex on an interval $I \subset \mathbf{R}$. Then at every interior point $a$ of $I$ the function $f$ is continuous, has finite right and left derivatives, and $f'_g(a) \leq f'_d(a)$.

Indeed, for $x \in I$ and $x > a$ the function $x \mapsto \frac{f(x) - f(a)}{x - a}$ is increasing (prop. 5) and bounded below, since if $y < a$ and $y \in I$ we have

(5)

by prop. 5; this function therefore has a finite right limit at the point $a$; in other words, $f'_d(a)$ exists and is finite; further, letting $x$ approach $a$ ($x > a$) in (5), it follows that

(6)

for all $y < a$ belonging to I. In the same way one shows that $f'_g(a)$ exists and that

(7)

for $x \in I$ and $x > a$. On letting $x$ approach $a$ ($x > a$) in this last inequality we obtain $f'_g(a) \leq f'_d(a)$. The existence of the left and right derivatives at the point $a$ clearly ensures the continuity of $f$ at this point.