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Math Help - Need help

  1. #1
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    Need help

    Can anyone help me please, I dont really have an idea.

    Suppose f is bounded on a closed interval and there exists a partition P for which U(f;P) = L(f;P). Is f Riemman integrable on the closed inteval?

    Thank you
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Green03 View Post
    Can anyone help me please, I dont really have an idea.

    Suppose f is bounded on a closed interval and there exists a partition P for which U(f;P) = L(f;P). Is f Riemman integrable on the closed inteval?

    Thank you
    for any partition P of [a,b],
    L(f;P) =< U(f;P), L(f) =< U(f)
    and
    L(f;P) =< L(f), U(f) =< U(f;P)

    this means that
    U(f) - L(f) =< U(f;P) - L(f;P)..

    if epsilon is any positive real such that there exists a partition P for which U(f;P) = L(f;P) (assumption) implying U(f;P) - L(f;P) < epsilon,

    then 0 =< U(f) - L(f) < epsilon, which implies U(f) - L(f) = 0.
    hence U(f) = L(f), therefore f is Riemann integrable on [a,b].QED
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