Continuity of functions and composition

Question

Let $f:X \rightarrow Y$, $g:Y \rightarrow Z$ be functions such that

1)$f$ is onto,

2)$f$ and $g \circ f$ are continuous.

Under what minimal requirements $g$ is continuous?

I am only aware of Theorem 1 in Kellum K.R., Rosen H. (1992). Compositions of continuous functions and connected functions.

The topic is related to this question.

Thank you.

Answer 1

If $f$ is also an open map, then for an open $U\subseteq Z$, we have $$g^{-1}(U)=f(f^{-1}(g^{-1}(U))) =f((g\circ f)^{-1}(U))$$ is open. (The first equality holds because of surjectivity.)