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I have to prove the following:

Let $f: \mathbb{R^2}\to \mathbb{R}$ such that $f_x:y\to f(x,y)$ is Borel measurable for all $x\in\mathbb{R}$ and that $f^y:x\to f(x,y)$ is continuous for all $y\in\mathbb{R}$. Prove that $f$ is Borel measurable.

What I have tried to do is to find a sequence of functions $f_n(x,y)$ s.t for a fixed $y$ $f_n(.,y)$ is a linear approximation of $f(.,y)$..

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By the continuity of $f^y$ we have $$f(x,y) = \lim_{n \to \infty}f(\lfloor nx \rfloor / n, y).$$ By the measurability of $f^x$, we see that $f$ is the pointwise limit of a sequence of Borel measurable functions, and hence is itself Borel measurable.

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  • $\begingroup$ how is $f(\lfloor nx \rfloor/n, y)$ borel? $\endgroup$ – Airdish May 19 at 23:08

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