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Suppose $f$(x,y) is a function on $\mathbb{R}^{2}$ that is separately continuous: for each fixed variable, $f$ is continuous in the other variable. Prove that $f$ is measurable on $\mathbb{R}^{2}$.

There is also a hint: Approximate $f$ in the variable $x$ by piecewise-linear functions $f_n$ so that $f_n$ $\rightarrow$ $f$ pointwise.

I don't get how to prove the problem via this hint, or is there any other approach?

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    It is actually sufficient that the function is continuous in one coordinate and measurable in the other. Such functions are known as *Caratheodory functions*.2012-12-28

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Hint: take $f_{m,n}(x,y):=f\left(\frac{\lfloor mx\rfloor}n,\frac{\lfloor ny\rfloor}n\right)$, where $\lfloor t\rfloor$ is the greatest integer lower than $t$. As $f_{m,n}$ takes only countably many values, it's measurable.

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    why does $f_{m,n}$ converges to $f$, and how is continuity used here?2012-12-28
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    For a fixed $m$, $f_{m,n}(x,y)$ converges to $f\left(\frac{\lfloor mx\rfloor}m,y\right)$ as the map $t\mapsto f\left(\frac{\lfloor mx\rfloor}m,t\right)$ is continuous. Do the same taking the limit with respect to $m$.2012-12-28
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    @DavideGiraudo I still have difficulty understanding the convergence. Can you help on my post http://math.stackexchange.com/questions/1388414/show-lim-m-to-infty-n-to-infty-f-frac-left-lfloor-mx-right-rfloo? Thank you!2015-08-08
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    Wait, so this answer does not work, if I understand the other question correctly?2018-07-14