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Math Help - Riemann Integral

  1. #1
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    Riemann Integral

    Suppose i have f(x)=0 for  x \neq \frac{1}{n},n=1,2,3,..... and  f(\frac{1}{n})=1 .
    how to show that f is Riemann Integrable and the riemann sum is 0 by using definition?
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  2. #2
    Super Member girdav's Avatar
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    Re: Riemann Integral

    Fix \varepsilon>0, and n_0 such that n_0^{-1}\leq \varepsilon. Put s_2(x):=\begin{cases} 1&\mbox{ if } x\leq n_0^{-1}\\0&\mbox{ otherwise}\end{cases} and s_1(x)=0. Choose a good subdivision and you are done.
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  3. #3
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    Re: Riemann Integral

    Quote Originally Posted by girdav View Post
    Fix \varepsilon>0, and n_0 such that n_0^{-1}\leq \varepsilon. Put s_2(x):=\begin{cases} 1&\mbox{ if } x\leq n_0^{-1}\\0&\mbox{ otherwise}\end{cases} and s_1(x)=0. Choose a good subdivision and you are done.
    Why we construct s_2 (x) and s_1 (x)? Can it related to f?
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  4. #4
    Super Member girdav's Avatar
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    Re: Riemann Integral

    It's just a definition. s_1 and s_2 are step functions. Using the subdivision 0<n_0^{-1}<(n_0-1)^{-1}<\ldots< 2^{-1}<1, you can check the definition of Riemann integrable.
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