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Difficult integration question

  1. Sep 17, 2007 #1
    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. jcsd
  3. Sep 18, 2007 #2
    No one has any suggestions? :cry:
     
  4. Jan 9, 2009 #3
    The first condition of this question is similar to that of Drichilet function. Break the condition further from this.
     
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