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Math Help - Riemann Sums and Riemann Integrals

  1. #1
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    Riemann Sums and Riemann Integrals

    The problem is: Prove that if f is continuous at x0, an element of [a,b], and f(x0)=/ 0, then the lower intergral from a to b of the absolute value of f(x) is >0.
    I am supposed to use the Sign Preserving Property, but I don't see how to set this up to make it work. I can see that I have to prove that either the abs. value of the inf{L(f,P): P is a partition of [a,b]} or the abs. value of the sup {L(f,P): P is a partition of [a,b]} >0 but I don't see the beginning steps of it.
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  2. #2
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    f is continous on [a.b]
    The problem also says f(x) not = 0 for all x in [a,b]
    Therefore, f(x)<0 on [a,b] or f(x)>0 on [a,b]
    Because if not, then by Intermediate value theorem there is a point such that f(x)=0 which we assumed was false.
    Therefore,
    |f(x)|>0
    Thus, by the inequalities of integration we have,
    INTEGRAL (a,b) |f(x)|dx>0
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