does composition of maps is smooth and one map is smooth imply the other is also smooth?

Question

If $f\circ g$ is smooth and $f$ is smooth, does it follow that $g$ is smooth? Note that I cannot simply take the inverse of $f$. Do I have to use implicit function theorem?

Answer 1

A composition of functions can smooth out irregular behaviour of either partner. If $f$ is constant, you cannot say anything about $g$. If $f$ is locally invertible, however (e.g. a local diffeomorphism), and $g$ is continuous, then the smoothness of the composition implies the smoothness of $g$.