So the problem is:
Let $$ f(x,y): [a,b] \times [c,d] \to \mathbf{R} $$ with type $$ f(x,y) = \begin{cases} 1, (x,y) \in (\mathbf{Q} \times \mathbf{Q}) \cap ([a,b] \times [c,d])\\ 0, \text{ else } \end{cases} $$ Prove that $f$ is not integrable in its domain.
So my try at an answer was by simply taking (letting $R = [x_{i-1},x_{i}] \times [y_{j-1},y_{j}]$ a random cut) $$ L(P,f) = \sum\sum m_{ij}E_{ij} $$ where $$ m_{ij} = \inf(f(x,y):(x,y) \in R)$$ so $$ L(P,f) = \sum\sum 0 E_{ij} = 0 $$ and by that same token $$ U(P,f) = \sum\sum 1 E_{ij} = E = 1 $$ and therefore taking into account the Raymond Criterion we get $$U(P,f) - L(P,f) < \epsilon$$ for $\epsilon = 1/2$ $$1 < 1/2$$ which is a contradiction.
Is my answer correct? What I think I lack is care for the various cases of the domain (if (x,y) is (rational,irrational) or (irrational,rational) or (irrational,irrational)). If they need to be mentioned, how would that need to be written?
Thanks in advance.