Suppose $f$ and $g$ are functions from $[0,\infty) \rightarrow [0,\infty)$. We are given that $f$ is non-increasing and differentiable, and $f\circ g$ is differentiable. Then under what additional conditions is it true that $g$ is differentiable?
You need some additional condition because the conclusion is not true if $f$ is a constant in any interval. Is the condition that $f' <0 $ sufficient?