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Integrability of f^2

  1. May 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Let f be positive and bounded over [a,b]. If f^2 is integrable over [a,b], then show that f is as well.

    3. The attempt at a solution

    I'm just trying to use the fact that the upper and lower sums of f^2 over a partition P are arbitrarily close, and then somehow find an upper bound for the difference of the upper and lower sums of f based on that. I've tried separating it into cases, where the max and min of f on an interval are >=1, <1, etc. but it hasn't really led anywhere. If anyone could give me some sort of hint, i'd really appreciate it =)
     
  2. jcsd
  3. May 15, 2008 #2

    Dick

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    It's fairly easy if you know f(x)>=m for m>0 isn't it? Since f(x1)^2-f(x2)^2=(f(x1)-f(x2))*(f(x1)+f(x2)) and you know f(x1)+f(x2)>=2m. So f(x1)^2-f(x2)^2>=(f(x1)-f(x2))*2m. Now let m approach zero. Why can you ignore the part of the sums coming from f(x)<m? Remember the interval of integration is [a,b]. It's finite.
     
    Last edited: May 15, 2008
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