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Absolute value of a function integrable?

  1. Nov 23, 2008 #1
    this is the question,
    Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.)

    I know you have to use the upper and lower bounds to prove this statement but i don't know where to start?

    Thanks
     
    Last edited: Nov 23, 2008
  2. jcsd
  3. Nov 23, 2008 #2

    lurflurf

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    Homework Helper

    let eps>0
    H(x):={y!=x| |f(x)-f(y)|<eps/(b-a)}
    Union[H(x)|x in [a,b]]
    is an open cover (by continuity of f) of [a,b] a compact set so we may chose a finite subcover
    is P is any partition at least as fine as the open cover will have
    U-L<eps
    qed
     
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