Consider the function $f$ defined by $$ f(x,y)= \begin{cases} 1 \iff (x,y) =\left(\frac p {2^n},\frac q {2^n}\right): (p,q,n) \in \Bbb N^3, 0<p,q<2^n \\0 \iff (x,y) \neq \left(\frac p {2^n},\frac q {2^n}\right) \end{cases} $$ I need to prove that the iterated integral on the rectangle $[0,1]\times[0,1]$ is zero, i.e
$$\int_0^1\int_0^1 f(x,y)dydx=\int_0^1\int_0^1 f(x,y)dxdy=0$$
The question was asked before, but only a hint is given (https://math.stackexchange.com/q/1035785)
What does it mean "not (relatively) dense" in $[0,1]$?
I am considering the function
$$F(x) = \int_{0}^1f(x,y)dy$$
which is evidently $0$ if $x \ne \frac{p}{2^n}$ and I am trying to show is $0$ everywhere.
Thanks a lot for the help.