Let $f:\mathbb{R} \to \mathbb{R}$ be a convex function. How many cusps (sharp corners) could it have? Are them numerable or not?
I could say that they are infinite and numerable, thinking to a function like $f(x)=\arcsin\bigg(\frac{\sin x}{\sqrt{2} - \cos x}\bigg)$, which is not convex. If I am right, how to prove it?