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Math Help - Bounded functions

  1. #1
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    Bounded functions

    Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
    by

    F(x)= \int_a^x f(t) dt

    ii) Prove that F is bounded.

    F is bounded if |F'|<=f yes? how do we know f is bounded?

    thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
    by

    F(x)= \int_a^x f(t) dt

    ii) Prove that F is bounded.

    F is bounded if |F'|<=f yes? how do we know f is bounded?

    thanks
    Since :

    F(x+h) =\int^{x+h}_{a}f(x)dx =\int^{x}_{a}f(x)dx+\int^{x+h}_{x}f(x)dx =F(x)+\int^{x+h}_{x}f(x)dx = F(x) +hf(c) ,where x<c<x+h, by the mean value theorem of integrals.

    Hence lim_{h\to 0}F(x+h) = F(x)

    Hence F continuous on [a,b] ,thus bounded on [a,b]
    Last edited by xalk; March 18th 2010 at 07:26 AM. Reason: not complete
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