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Math Help - iintegrability question of irrational

  1. #1
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    iintegrability question of irrational

    3)b)
    there is a continues and positive f in [0,1]
    prove that g(x) is not integrabile in [0,1]
    g(x)=f(x) for rational x
    g(x)=-f(x) for irational x
    S(p)=\sum_{i=1}^{n}M_{i}(x_{i}-x_{i-1})


    why its not integrible?
    because f the endless points of disconinuety?

    my prof showed a function
    g(x)=1 for rational x
    g(x)=1/x for irational x
    and he said that it is integrible

    i dont know why the original is not integrible
    ?
    Last edited by transgalactic; September 6th 2011 at 02:04 PM.
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  2. #2
    Super Member TheChaz's Avatar
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    Re: iintegrability question of irrational

    Regardless of the partition, the upper sums will be...
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  3. #3
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    Re: iintegrability question of irrational

    the upper sum will be
    S(p)=\sum_{i=1}^{n}Sup(g([0,1]))(x_{i}-x_{i-1})=\sum_{i=1}^{n}f(x)(x_{i}-x_{i-1})
    the lower sum
    will be
    S(p)=\sum_{i=1}^{n}inf(g([0,1]))(x_{i}-x_{i-1})=\sum_{i=1}^{n}(-f(x))(x_{i}-x_{i-1})

    i dont know if the supremum is actually is f(x) because its not an actual number.
    same thing for the infinum
    i just guessed because this is the only thing we've got.

    but in order to prove that its not integrabile
    their subtraction shoudnld be lowe then epsilon
    dont know how to show that

    ??
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  4. #4
    Super Member TheChaz's Avatar
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    Re: iintegrability question of irrational

    I should have hinted at the lower sum. On any interval, there will be a rational number, so the lower sums will be 1.
    The uppers will not be one!
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    Re: iintegrability question of irrational

    how you got to the conclution that f(x)=1
    ?

    oohh i know that
    my prof sayed that we are working only on darbu integral

    what you are saying is lebeg integral
    and its beyong the scope of my course
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  6. #6
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    Re: iintegrability question of irrational

    Quote Originally Posted by transgalactic View Post
    3)b)
    there is a continues and positive f in [0,1]
    prove that g(x) is not integrable in [0,1]
    g(x)=f(x) for rational x
    g(x)=-f(x) for irational x
    why its not integrible?
    because f the endless points of disconinuety?
    Assuming that you mean Riemann Integration.
    Is it possible that g is continuous at any point in [0,1]~?
    Do you have a theorem on the cardinality of the set of discontinuities of an integrable function?
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  7. #7
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    Re: iintegrability question of irrational

    i just remmember the words of my prof that said that in our couse dereclet function is not integrible.
    but in other courses it is integrible and the integral is 1

    so when TheChaz said that the result is 1
    i got remmemebered the above remark.

    regradring your question:
    yes it is possible if f(x)=0,but i was given that its possitive so i cannot happen
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