The following question is asked in the book of Analysis On Manifolds by Munkres, and given at page 103 as question 3.
$Q=A\times B$,where $A$ is a rectangle in $\mathbb{R^k}$ and $B$ is a rectangle in $\mathbb{R^n}$.
- Give an example where $\int_{Q}f$ exists and one of the iterated integral
$$\int_{x\in A}\int_{y\in B} f(x,y) \; \text{and} \; \int_{y\in B}\int_{x\in A} f(x,y)$$ exists, but the other does not.
- Find an example where both the iterated integrals of 1. exist, but the integral $\int_Q f$ does not. [Hint: One approach is to find a subset $S$ of $Q$ whose closure equals $Q$, such that $S$ contains at most one point on each vertical line and at most one point on each horizontal line.]
I'm having difficulty coming up with examples for the second case using the hint.
For the first one, I found $f(x,y)= 1/n$ if $y$ is rational and $x=m/n$, where $(m,n)=1$.
However, I have no clue about the second one, and what the hint even means.
I'd appreciate it if anyone can help me with this problem.