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Math Help - need a counterexample for Riemann Integrable functions problem

  1. #1
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    need a counterexample for Riemann Integrable functions problem

    I need to show that the following statement is false using a counterexample, problem is I cant come up with one!

    Let f,h be Riemann integrable on the interval [a,b].
    Let f(x)<=g(x)<=h(x) for all x in [a,b].
    Then g is Riemann integrable.

    any suggestions?
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    I need to show that the following statement is false using a counterexample, problem is I cant come up with one!

    Let f,h be Riemann integrable on the interval [a,b].
    Let f(x)<=g(x)<=h(x) for all x in [a,b].
    Then g is Riemann integrable.

    any suggestions?
    on the interval I=[0,1] define f(x)=0, \ h(x) = 1, \ \forall x \in I. now define g: I \longrightarrow \mathbb{R} by g(x)= 1, \ \forall x \in I \cap \mathbb{Q}, and g(x)=0 otherwise.
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