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Math Help - Finding if a function is Riemann Integrable

  1. #1
    xyz
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    Finding if a function is Riemann Integrable

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    Quote Originally Posted by xyz View Post

    Well, going with the hint you could argue that since f(x) is continuous in [\theta,1], for any \theta>0, then f is uniformly continuous there, so \forall\,\, \epsilon > 0\,\,\exists\,\,\delta>0\,\,s.t.\,\,|x-y|<\delta\,\Longrightarrow |f(x)-f(x)|<\epsilon , and then: taking a partition P with n points of [\theta,1] s.t. \max \{x_i-x_{i-1} /\,[x_{i-1},x_i ]\in P\}<\frac{\delta}{n\epsilon}, we get U(f,P)-L(f,P)<\epsilon.
    Of course, we still need to deal with [0,\theta), but this is not problem: the difference \theta-0=\theta. can be made as little as wanted.

    Now, not going with the hint is much simpler and shorter, but perhaps you guys haven't yet studied this: f is Riemann integrable there since it is bounded and the set of points of discontinuities of f in [0,1] (which is one single point) has (Borel-Lebesgue) measure zero.

    Tonio
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  3. #3
    xyz
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    As seen above is the graph of the function that has to be proven, it is riemann integrable. So this is how i answered. Please let me know if it is correct --


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