Corollary

Let $f$ be a finite real function, continuous and twice differentiable on an interval $I \subset \mathbf{R}$; for $f$ to be convex on $I$ it is necessary and sufficient that $f''(x) \geqslant 0$ for all $x \in I$; for $f$ to be strictly convex on $I$ it is necessary and sufficient that the previous condition be satisfied and further that the set of points $x \in I$ where $f''(x) > 0$ be dense in $I$.

This follows immediately from the preceding proposition, and from the corollary at I, p. 14.