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Bounded variation function

  1. Jan 16, 2010 #1
    1. The problem statement, all variables and given/known data

    f is of bounded variation on [a;b] if there exist a number K such that

    [tex]\sum[/tex][tex]^{n}_{k=1}[/tex]|f(ak)-f(ak-1)| [tex]\leq[/tex]K

    a=a_0<a_1<...<a_n=b; the smallest K is the total variation of f

    I need to prove that
    1) if f is of bounded variation on [a;b] then it is bounded on [a;b]
    2) if f is of bounded variation on [a;b], then it is integrable on [a;b]

    2. The attempt at a solution

    i thought of using triangle inequation such that
    0<=|f(b)-f(a)|<=[tex]\sum[/tex][tex]^{n}_{k=1}[/tex]|f(ak)-f(ak-1)| [tex]\leq[/tex]K

    but im not really sure how to prove the two statements
    any help is really appreciated
  2. jcsd
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