Let $f,g:\Bbb R\to\Bbb R$ be real variable continuous functions and define $h:=f+g$; suppose $h$ non-constant.
If $h$ is differentiable at some $x_0\in\Bbb R$, does the same holds for both $f$ and $g$? Or there exist a couple of functions $f,g$ not differentiable at $x_0$ such that their sum does?
Maybe is trivial but I can't find counterexamples.
In what case is this true?