# differentiability of functions under composition

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?

It is sufficient, because $f'<0$ implies that $f$ is strictly decreasing, hence bijective onto its image (which must be an interval). Moreover, since $f'$ has no zeors, it holds $$(f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}$$ so $g=f^{-1}\circ f\circ g$ is differentiable.