Definition 1

We say that a finite numerical function $f$, defined on an interval $I \subset \mathbf{R}$, is convex on $I$ if, no matter what the points $x, x'$ of $I$, ($x < x'$), every point $M_z$ of the graph $G$ of $f$ such that $x \leq z \leq x'$ lies below the segment $M_xM_{x'}$ (or, what comes to the same, if every point of this segment lies above $G$) (fig. 2).

!Graph showing points M_x, M_z, M_{x'}, and curve G

Fig. 2

Taking account of the parametric representation of a segment (Gen. Top., VI, p. 35), the condition for $f$ to be convex on I is that one has the inequality

for each pair of points $(x, x')$ of I and every $\lambda \in [0, 1]$.

Definition 1 is equivalent to the following: the set of points in $\mathbf{R}^2$ lying above the graph G of $f$ is convex. Indeed, this condition is clearly sufficient for $f$ to be convex on I; it is also necessary, for if $f$ is convex on I, and if $(x, y), (x', y')$ are two points lying above G, then one has $y \geq f(x),\ y' \geq f(x')$, from which, for $0 \leq \lambda \leq 1$,

by (1), which shows that every point of the segment with endpoints $(x, y)$ and $(x', y')$ lies above G.