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Counterexamples needed for integration question

  1. Aug 5, 2007 #1
    1. The problem statement, all variables and given/known data

    The original question required me to show that for f(x) >= 0 for all x, f continuous, where the integral (from a to b) of f =0, that f(x) = 0 for all x in [a,b]. I did that, using a proof by contradiction.

    Second part of the question requires me to show that the two hypotheses (f(x) being >= 0 and f being continuous) were required.

    2. Relevant equations



    3. The attempt at a solution

    I think counterexamples would show this, but can't figure out what would make a counterexample. Do I need to take f(x)<0 for f continuous and show that its integral can't equal zero? Similarly, take f(x)>=0 but not continuous and show that its integral also can't equal zero?
     
  2. jcsd
  3. Aug 5, 2007 #2

    morphism

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    Almost. You want the integral to be zero. And for the first bit, you don't really need f(x)<0 for all x, just f(x0)<0 for some x0 in [a,b]. (Actually if f(x)<0 for all x, it won't work, because the integral won't be zero.)
     
    Last edited: Aug 6, 2007
  4. Aug 5, 2007 #3
    so would taking f(x) = x^3, which is continuous, be a suitable counterexample for the first assumption?

    I can't think of a function that is always positive and isn't continuous, though.
     
  5. Aug 5, 2007 #4

    morphism

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    Yup. That works provided you integrate on some interval symmetric about 0, like [-1,1].

    How about

    [itex]
    f(x)=\begin{cases}
    0 &\text{if } x \neq 0\\
    1 &\text{if } x = 0
    \end{cases}
    [/tex]

    for the second one?
     
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