Q: Integration of a function of two variables How can we check the integrability of $f$ defined on $[0,1] \times [0,1]$ as
$f(x,y)=$\begin{cases} 
      0 & x=\frac{1}{2},y \in \mathbb Q \\
      1 & x=\frac{1}{2},y \in \mathbb Q^c \\
      0 & x \in \mathbb Q, y=\frac{1}{2} \\
      1 & x\in \mathbb Q^c, y=\frac{1}{2} \\
      0  elsewhere
    \end{cases}
I know that $f$ is reimann integrable if the set of discontinuities has measure zero but I think i am unable to see things clearly here. The problem says it is integral on $[0,1] \times [0,1]$ but the integral $\int_{0}^{1}f(\frac{1}{2},y)dy$ doesn't exists. Is that even possible?
Any hints please? for $\int_{0}^{1}f(\frac{1}{2},y)dy$ I see that the whole real line is the discontinuity of $f$, hence it can't be integrable. What about the whole domain?
