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Riemann Integrable

  1. Nov 14, 2008 #1
    Here is the classic Dirichlet function:

    Let, for x ∈ [0, 1],
    f (x) =1 /q if x = p /q, p,q in Z

    or 0 if x is irrational.

    Show that f (x) is Riemann integrable and give the value of the integral.


    Is this actually true?
     
    Last edited: Nov 15, 2008
  2. jcsd
  3. Nov 15, 2008 #2
  4. Nov 15, 2008 #3

    morphism

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    Your function isn't defined at zero, but let's ignore that. Where does Wikipedia say that it's not integrable? It is in fact Riemann integrable. Its set of points of discontinuity has measure zero.
     
  5. Nov 15, 2008 #4

    mathman

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    It is Lebesgue integrable, but not Riemann. For Riemann, you need continuity except at a countable set.
     
  6. Nov 15, 2008 #5

    Hurkyl

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    That's a sufficient condition, not a necessary condition. The necessary and sufficient condition is that the function be discontinuous on a set of (Lesbegue) measure zero. (This function satisfies that condition)

    I think the problem isn't too hard if you just start writing down Riemann sums, and use approximations to simplify things.
     
  7. Nov 15, 2008 #6

    morphism

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    Ditto Hurkyl -- and in any case, this function is actually continuous everywhere except at a countable set (namely the rationals in [0,1]).
     
  8. Nov 15, 2008 #7

    HallsofIvy

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    What Wikipedia says is
    That is NOT the "Dirichlet function" the OP was talking about. The particular Dirichlet function Wikipedia is referring to (There are several) is discontinuous everywhere.
     
  9. Nov 15, 2008 #8

    rbj

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    but for x > 0, the given function is always less than the Dirichlet function (where not zero, the given function 1/q is less than 1). (it appears to be periodic with any period that is 1/q for integer q.) and we know that the Dirichlet function has Lebesque integral of zero and if this is Riemann integrable, i think the two integrals (over the same limits) has to be the same, no?
     
  10. Nov 15, 2008 #9

    morphism

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    I don't see what you're objecting to. Are you arguing that the given function isn't Riemann integrable?
     
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