Let $I$ be an interval and $f:I\rightarrow\mathbb{R}$ be a convex function. If $x_o\in\text{Int{I}}$, then $f_R^'(x_0)$ and $f_L^'(x_0)$ both exist.
I'm a little stumped here -- I know I'm supposed to use the fact that $f$ is a convex function so that given an interval $I$, where $a,b,c\in I$ we have an $\alpha\in (0,1)$ s.t. $b=\alpha a + (1-\alpha)c$.