Assume we have a function class $F$ containing bivariate functions $f(x,y)\; (f: \mathcal{X} \times \mathcal{Y} \to \mathbb{R})\ $ that are continuously differentiable with respect to each argument. The functions are strictly increasing in their first argument.

I've read a statement that when $g,h \in F$ satisfy

$$ g(x_1, y_1) < g(x_2, y_2) \Rightarrow h(x_1, y_1) < h(x_2, y_2) $$

for arbitrary $(x_1, y_1), (x_2, y_2)$ then there exists a strictly increasing function $m: \mathbb{R} \to \mathbb{R}$ such that $h = m \circ g$.

Unfortunately the statement is without proof and I have trouble to verify it.

It is clear that by strict monotonicity of the first argument there is an "inverse" function such that $g^{-1}(g(x,y),y)=x$ for a given $y$, but I am not able to arrive at the statement from there on.


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