Exercise 8, § 4

Let $f$ be a finite real function, upper semi-continuous on an open interval $I \subset \mathbf{R}$. Then $f$ is convex if and only if $\limsup_{h \to 0, h \neq 0} \frac{f(x+h) + f(x-h) - 2f(x)}{h^2} \geq 0$ for all $x \in I$. (First show that, for all $\varepsilon > 0$ the function $f(x) + \varepsilon x^2$ is convex, using prop. 9 of I, p. 31.)