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Math Help - Integrable functions.

  1. #1
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    Integrable functions.

    Hi,
    I need to find a function f, which is not integrable in [0,1], but the function |f| is integrable in that interval [0,1].

    can someone please help me with this?
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  2. #2
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    Quote Originally Posted by Boaz View Post
    Hi,
    I need to find a function f, which is not integrable in [0,1], but the function |f| is integrable in that interval [0,1].
    f(x) = \left\{ {\begin{array}{rl}   {1,} & {x \in \left[ {0,1} \right] \cap \mathbb{Q}}  \\   { - 1,} & {x \in \left[ {0,1} \right]\backslash \mathbb{Q}}  \\ \end{array} } \right.

    What can you do with that and why?
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  3. #3
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    Quote Originally Posted by Plato View Post
    f(x) = \left\{ {\begin{array}{rl}   {1,} & {x \in \left[ {0,1} \right] \cap \mathbb{Q}}  \\   { - 1,} & {x \in \left[ {0,1} \right]\backslash \mathbb{Q}}  \\ \end{array} } \right.

    What can you do with that and why?
    Im not sure i understand why f isnt integrable..is it because the upper and the lower sums are very "far" from each other in every interval of the partition?
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  4. #4
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    Quote Originally Posted by Boaz View Post
    Im not sure i understand why f isnt integrable..is it because the upper and the lower sums are very "far" from each other in every interval of the partition?
    Well, yes. That function is nowhere continuous.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Well, yes. That function is nowhere continuous.
    oh right, I forgot about that!
    Thank you very much!
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