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Math Help - show function is not integrable

  1. #1
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    show function is not integrable

    Let [0,1] ----> R be given by f(x) = 1 if x is rational, -1 if x is rational.

    Decide whether f is integrable.



    I know the function is not integrable, but I don't have any idea how to go about showing it? Been looking through some Riemann integral lecture notes but don't understand it at all!

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by hunkydory19 View Post
    Let [0,1] ----> R be given by f(x) = 1 if x is rational, -1 if x is rational.

    Decide whether f is integrable.
    Let P = \{x_0 < x_1 < ... <x_n] be a partition. Then the upper Darboux sum of f with respect to P is 1 (why?). Thus, the infimum of all upper Darboux sums is 1. While the lower Darboux sum of f with respect to P is 0 (why?). Thus, the supremum of all lower Darboux sums is 0. This means the upper integral does not agree with the lower integral. Thus, the (Dirichlet) function is not integrable on [0,1].
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  3. #3
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    Thanks so much for the reply, but for the lower Darboux sum I don't understand why it is 0 and not -1...
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    Quote Originally Posted by hunkydory19 View Post
    Thanks so much for the reply, but for the lower Darboux sum I don't understand why it is 0 and not -1...
    It is -1, I made a mistake.
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