# Difficult integration question

1. Sep 17, 2007

### regularngon

1. The problem statement, all variables and given/known data
Show that the function f : [0,1] × [0,1] → R given by

f(x,y) =

{ 0 if x is irrational, or x is rational and y is irrational
{ 1/q if x is rational, y = p/q with gcd(p,q) = 1

Is integrable and compute the integral.

2. Relevant equations

3. The attempt at a solution

I know I have to use the fact that Riemann integrability is equivalent to the fact that for every E > 0 there exists a partition P such that U(f,P) - L(f,P) < E.

Due to the density of the rationals in the reals, we are always going to have L(f,P) = 0. So I just have to find a partition P such that U(f,P) < E. So I'm quite sure that I'm going to have to use the infinite sum of 1/2^n. However, I'm quite stuck on figuring out a valid partition. The more I think, the harder finding this partition seems to be :(

Any suggestions? Thanks.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 18, 2007

### regularngon

No one has any suggestions?

3. Jan 9, 2009