The right derivative of a convex function $f$ is right continuous $\iff$ $f$ is differentiable Let be $f$ a convex function defined on an open set. We know from theory that $f'_{+},f'_{-}$ both exist not decreasing. **Claim :** The right derivative of a convex function $f$ is right continuous $\iff$ $f$ is differentiable. I think $[\Leftarrow]$ follows directly from definition because if $f'$ exists, both left and right limits exist finite and has to be coincident $f'_{+} = f'_{-} = l \in \mathbb{R}$ so $f'_{+}$ is continuous. I'm stuck with $[\Rightarrow]$. I was unable to approach the problem by any angle. Any help, hint or solution would be appreciated.

This is not true. Let $f(x)=0$ for $x<0$ and $f(x)=x$ for $x \geq 0$. Then $f'(x+)$ is right continuous but $f$ is not differentiable at $0$.