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Math Help - continuity question

  1. #1
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    Question continuity question

    If f is a function defined on [a,b] which is not continous,then L(P,f)> U(P,f) for any partition P of [a,b].State true or false. Give reason?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by puneet View Post
    If f is a function defined on [a,b] which is not continous,then L(P,f)> U(P,f) for any partition P of [a,b].State true or false. Give reason?
    false..
    Theorem: Let f: [a,b] \rightarrow \mathbb{R} be a bounded function. Then L(f) and U(f) exist; and L(P,f) \leq L(f) \leq U(f) \leq U(P,f)..

    the theorem does not have any condition on f to be continuous or not, thus it holds for any bounded function f: [a,b] \rightarrow \mathbb{R}


    if you have not seen that theorem, you may use a counter-example (Modified Dirichlet Function).
    f(x) = 1 if x\in \mathbb{Q} \cap [0,1]
    f(x) = 0 if x\in \mathbb{Q}^\prime \cap [0,1]

    and show that L(P,f) < U(P,f)
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