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Math Help - do i use integration?

  1. #1
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    do i use integration?

    if f is bounded on [0,1] and satisfies the condition |f(a) -f(b)| is less than
    |a-b| for all a, b on [0,1] , prove that f is Riemann integrable on [0,1]


    I have always been slightly confused by Riemann.
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  2. #2
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    the condition |f(a) -f(b)| is less than
    |a-b| for all a, b on [0,1]
    Does this not show that f is countinous of [0,1]? Now use the integration theorem which states that every closed countinous function is Riemann integrable.
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  3. #3
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    Yes. it does show f is continuous. But I don't know how to show that it is Riemann integrable. I don't think I ever grasped that when taught.

    Can you explain some more?
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  4. #4
    TD!
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    Bounded, continuous function are Riemann integrable, haven't you seen that?
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  5. #5
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    Quote Originally Posted by TD!
    Bounded, continuous functions
    Perhaps, the use of "bounded" was unnesseray because a countinous function on a closed interval is bounded.
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  6. #6
    TD!
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    Then I would've needed to include 'on a closed interval'
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  7. #7
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    Quote Originally Posted by TD!
    Then I would've needed to include 'on a closed interval'
    Good point.


    --
    What about,
    f(x)=x boundend and countinous on [0,\infty] but not riemann integrable. Thus, you need to include 'on a closed interval' anyway.
    Last edited by ThePerfectHacker; June 4th 2006 at 10:29 AM.
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