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Thread: Integrating Functions with Discontinuities

  1. #1
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    Integrating Functions with Discontinuities

    Hey, I've got a real analysis question here. Any help would be greatly appreciated:

    Let f(x) = {1 if x = 1/n for some n Є N
    {0 otherwise

    Show that f is integrable on [0,1] and compute
    $\displaystyle
    \int_1^0 {f}
    $
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  2. #2
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    Quote Originally Posted by ajj86 View Post
    Hey, I've got a real analysis question here. Any help would be greatly appreciated:

    Let f(x) = {1 if x = 1/n for some n Є N
    {0 otherwise

    Show that f is integrable on [0,1] and compute
    $\displaystyle
    \int_1^0 {f}
    $

    consider this partition let $\displaystyle \epsilon >0 $

    P\fix N such that $\displaystyle \frac{1}{N} < \frac{\epsilon}{4}$

    $\displaystyle p_n=\left( \frac{1}{n}-\frac{\epsilon}{2^{n+1}},\frac{1}{n}+\frac{\epsilo n}{2^{n+1}} \right)$

    for n=2 to n=N

    notice that $\displaystyle \sum_{n=2}^{N}\left(\frac{1}{n}+\frac{\epsilon}{2^ {n+1}}-\left( \frac{1}{n}-\frac{\epsilon}{2^{n+1}}\right) \right)=\sum_{n=2}^{N}\frac{\epsilon}{2^{n}}<\frac {\epsilon}{2}$

    See if you can put it together from here

    Good luck.
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