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Math Help - analysis problem

  1. #1
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    analysis problem

    Let f:[0,1] -> [0,1] be defined as follows:
    f(x)=1/n if x=m/n where n,m are integers, n is not zero, and m/n is irreducible.
    f(x)= 0 if x is irrational.
    Prove that f(x) is Riemann integrable on [0,1]

    please help me on this.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    Let f:[0,1] -> [0,1] be defined as follows:
    f(x)=1/n if x=m/n where n,m are integers, n is not zero, and m/n is irreducible.
    f(x)= 0 if x is irrational.
    Prove that f(x) is Riemann integrable on [0,1]

    please help me on this.
    What do you have to use? That function, the "modified Dirchlet function" (the Dirichlet function itself is 0 for x rational, 1 for x irrational), can be proved to be continuous exactly on the irrational numbers and 0. That means that it is a bounded function whose set of discontinuities has measure 0. There is a theorem that says such a function is Riemann integrable.
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  3. #3
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    still having a problem

    Quote Originally Posted by HallsofIvy View Post
    What do you have to use? That function, the "modified Dirchlet function" (the Dirichlet function itself is 0 for x rational, 1 for x irrational), can be proved to be continuous exactly on the irrational numbers and 0. That means that it is a bounded function whose set of discontinuities has measure 0. There is a theorem that says such a function is Riemann integrable.
    thank you very much for the reply. but i am not really getting it. i ve never heard of Dirchelt and i dont understant what it means to say the set of discontinuities has measure 0. is there any other way? if so i would really appriciate it.
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