Corollary 2

Let $f$ be a finite real function defined on an open interval $I \subset \mathbf{R}$. If for every point $x \in I$ there is an open interval $J_x \subset I$ containing $x$ and such that the restriction of $f$ to $J_x$ is convex on $J_x$, then $f$ is convex on $I$.

It is clear that $f$ satisfies the criterion of prop. 9.