I have to show that a bounded differentiable convex function $f: \Bbb R \rightarrow \Bbb R $ is constant.
When a function $f$ is differentiable and convex, then I have a Theorem in my book that says, that:
$ f(y) \ge f(x) + f'(x)(y-x) \quad \forall x,y \; \in \Bbb R $
I'm stuck from here.