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Math Help - examining a riemann integral

  1. #1
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    examining a riemann integral

    Let g(x)=0 if x \in [0,1] is rational and g(x)=1/x if x \in [0,1] is irrational. Explain why g is not riemann integrable.
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    Let g(x)=0 if x \in [0,1] is rational and g(x)=1/x if x \in [0,1] is irrational. Explain why g is not riemann integrable.
    when taking dissections every interval contains rational and irrational points, so the lower sums are always equal to zero for every dissection but the upper sums are always greater than zero.

    Bobak

    edit: sorry that isn't a complete justification because the upper integral could could be arbitrary small ( i.e. < \epsilon \ \ \forall   \epsilon > 0 ) however it is easy to show that 1 is a lower bound on the upper integral.
    Last edited by bobak; April 1st 2009 at 06:29 AM.
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  3. #3
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    Also for a function to be Riemann integrable doesn't it have to be bounded? So g(x_n)\to\infty as x_n\to{0} and x_n is irrational.
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