Example

On the interval ]$0, +\infty$ [ the function $x^r$ ($r$ any real number) has a second derivative equal to $r(r-1)x^{-2}$; thus it is strictly convex if $r > 1$ or $r < 0$, and strictly concave if $0 < r < 1$. \*

In order to be able to formulate another criterion for convexity we make the following definition: given the graph $G$ of a finite real function defined on an interval $I \subset \mathbf{R}$ and an interior point $a$ of $I$, we shall say that a line $D$ passing through $M_a = (a, f(a))$ is locally above (resp. locally below) $G$ if there exists a neighbourhood $V \subset I$ of $a$ such that every point of $D$ contained in $V \times \mathbf{R}$ is above (resp. below) $G$; we shall say that $D$ is locally on $G$ at the point $M_a$ if there is a neighbourhood $V \subset I$ of $a$ such that the intersection of $D$ and $V \times \mathbf{R}$ is identical to that of $G$ and $V \times \mathbf{R}$ (in other words, if $D$ is simultaneously locally above and locally below $G$).