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Example of a non-integrable function f , such that |f| and f^2 are integrable?

  1. Feb 3, 2009 #1
    I'm looking for a function
    f:[a,b] -> R such that |f| and f2 are integrable on [a,b]
    any helps?
  2. jcsd
  3. Feb 3, 2009 #2
    To start with, f must be discontinuous (a continuous function on a closed interval is integrable). Can you think of a function f such that f is discontinuous, but |f| (and thus f2 = |f|2) is continuous?

    This is really an analysis question, so it really belongs in that forum. :)
    Last edited: Feb 4, 2009
  4. Feb 4, 2009 #3
    f(x)= 1 for x rational,
    -1 for x irational
  5. Feb 4, 2009 #4
    Don't just give him an answer. ;)

    (That's the exact same function I was thinking of, though.)
  6. Feb 4, 2009 #5
    I forgot to say: Only if you are dealing with newton or rieman integrals, otherwise f(x) is integrable.
  7. Feb 4, 2009 #6
    Oh I don't know why I didn't think of that!!
    Thank you so much to both of youuuu!
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