$f,g$ are real-valued, $f(x,y)=g(h_1(x),h_2(y))$, $f,h_i$ are continuous, $g$ is increasing, does $g$ must be continuous?

Question

Assumptions: $f,g$ are real-valued. $g:[0,1]^2\to\mathbb R$. Functions $h_1:X\to\mathbb [0,1]$ and $h_2:Y\to\mathbb [0,1]$ are surjective continuous. $X,Y$ are connected separable. $f$ is continuous,

$f(x,y)=g(h_1(x),h_2(y))$.

Questions: Does the strict increasingness of $g$ in all variables imply the continuity of $g$?

Backgrounds: It is known the composite of continuous functions is continuous. What about the other way around?

What we know: even without increasingness, $g$ is continuous if one of the followings hold:

1. X, Y are path-connected
2. X,Y are compact Hausdorff
3. $h_i$ are homomorphisms
4. $h_i$ are quotient maps

To prove the claim, we only need to show that if $g$ is continuous restricted to one direction, then $g$ is continuous. Also, since $g$ is increasing, the restricted function of $g$ can only contain a jump discontinuity. (If we allow essential discontinuity, a simple counterexample exists, this is why the increasingness is powerful here). I've been trying to construct a counterexample with jump discontinuity for a while, with no luck.