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Math Help - Riemann integrable

  1. #1
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    Riemann integrable

    Let
    f : [a, b]--->R be bounded. Prove or disprove: if f^{2} is Riemann integrable on [a, b] then f is Riemann integrable on [a, b].


    I have a set of theorems in my notes but I can't find the correct one to apply to the above problem.



    if f^{2} is Riemann integrable on [a, b] then f is has also Riemann integrable on [a,b] correct? How do i prove this?



    thanks for any help.
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  2. #2
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    For the first one, without knowing what theorems you have available, I don't see how we can assist. I would have used "a bounded function is integrable on [a, b] if and only if its set of points of discontinuity in [a, b] has measure 0". Do you have that?

    As for the second problem, consider f(x)= 1 if x is rational, -1 if x is irrational.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    For the first one, without knowing what theorems you have available, I don't see how we can assist. I would have used "a bounded function is integrable on [a, b] if and only if its set of points of discontinuity in [a, b] has measure 0". Do you have that?

    As for the second problem, consider f(x)= 1 if x is rational, -1 if x is irrational.
    Aren't the first and second questions the same?
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  4. #4
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    I have only question, the first question.

    I just wrote it the other way round in the second part.

    thanks
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