Can a non-trivial continuous function "undo" the discontinuities of another function?

Apologies for the unclear title, I have no idea if the property I'm looking for has a better name.

I'm wondering if there exists a pair of functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ such that :

1. $g$ is a bijection and is nowhere continuous (for an example, see this answer).
2. $f$ is continuous and not constant.
3. $f \circ g$ is continuous.