Corollary 1

For a real finite function $f$ defined on an open interval $I \subset \mathbf{R}$ and upper semi-continuous on $I$ to be convex on $I$ it is necessary and sufficient that for all $x \in I$ there should exist an $\varepsilon > 0$ such that the relation $|h| \leq \varepsilon$ entails

We have only to show that the condition is sufficient. Indeed, if at a point $M_a$ of the graph G of $f$ a line D is locally above G, then it is locally on G at this point; for, in the opposite case, for example, a point $M_{a+h}$ would be strictly below D, while a point $M_{a-h}$ would be below D: the mid-point of the segment $M_{a-h}M_{a+h}$ would thus be strictly above D (fig. 5), and, in virtue of the hypothesis, $M_a$ would a fortiori be strictly below D, which is absurd.

!A graph showing a convex function with points M_{a-h}, M_a, M_{a+h} and a tangent line labeled D.

Fig. 5