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Math Help - Integrability Proof

  1. #1
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    Integrability Proof

    Prove that the piecewise function f(x) = 1 , if x =1/2 and f(x) = 0, otherwise is integrable on the interval [0, 1] using the Darboux Method.


    I understand that the integral from x = 0 to x = 1/2 will evaluate to 0 and the integral from x = 1/2 to x = 1 also evaluates to 0.
    But how can I show this more formally? The issue arises at x = 1/2, how do I deal with that if using Darboux upper and lower integrals?
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  2. #2
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    Re: Integrability Proof

    When you divide [0,1] into sub-intervals, there will only be one sub-interval containing 1/2. The upper sum will be the length of this interval, the lower sum will be zero. The lower integral is obviously zero. The challenge is to prove that the upper integral is zero.

    It shouldn't be hard to find a partition of [0,1] for which the upper integral is less than \epsilon for any \epsilon>0. You can use this to prove that the upper integral is zero.

    - Hollywood
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  3. #3
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    Re: Integrability Proof

    Are you referring to the Cauchy criterion for integrability?
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  4. #4
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    Re: Integrability Proof

    I was thinking of the definition upper integral = lower integral, but the Cauchy criterion seems to work, too.

    - Hollywood
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