# Existence of a function representation

Let $f: [0,1]^2 \to R$ be an arbitrary continuous in both arguments and increasing in the first argument function, and let $h: [0,1]^2 \to [0,1]$ be some arbitrary function.

Does $\forall f,h$ there exists $g: R^2 \to R$ such that $\int_0^1f(x,h(z,y))dy=f(x,g(x,z))$ $\forall x,z$?

Thank you!!!

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For the integral to be correctly defined you must say something about the measurability of $h$. You ask nothing about the regularity of $g$. The relation you just wrote could pass as the definition of $g$ if you use the intermediate value theorem for $f(x,\cdot)$ –  Beni Bogosel Oct 31 '12 at 20:20