Let $f: \mathbb{R} \to [0, \infty)$ be strictly increasing and convex. Since $f$ is convex its right derivative at, say, zero exists and is finite.
Is it possible to construct an $f$ such that its right derivative is zero at some point?
Intuitively, this should not be possible since it would constitute a point where the tangent at $f$ (from the right) is flat and $f$ can not become smaller when we move to the left, i.e. we would either loose convexity or strictly increasing"ness". However, I cannot find a proper proof or counterexample.
Best Answer
Say $D_Rf$ is the right derivative. $f$ convex implies that $D_Rf$ is non-decreasing, while $f$ strictly increasing implies that $D_Rf\ge0$. So if $D_Rf(0)=0$ then $D_Rf(x)=0$ for all $x<0$, hence $f$ is not strictly increasing.