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Math Help - Real Analysis: Prove BV function bounded and integrable

  1. #1
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    Exclamation Real Analysis: Prove BV function bounded and integrable

    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

    \sum ^{n}_{k=1}|f(ak)-f(ak-1)| \leqK

    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)|<= \sum ^{n}_{k=1}|f(ak)-f(ak-1)| \leqK

    but im not really sure how to prove the two statements
    any help is really appreciated
    thanks
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  2. #2
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    Quote Originally Posted by kfdleb View Post
    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

    \sum ^{n}_{k=1}|f(ak)-f(ak-1)| \leqK

    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)|<= \sum ^{n}_{k=1}|f(ak)-f(ak-1)| \leqK

    but im not really sure how to prove the two statements
    any help is really appreciated
    thanks
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  3. #3
    Senior Member Dinkydoe's Avatar
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    On other sites they use  code but here tex = math
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kfdleb View Post
    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

    \sum ^{n}_{k=1}|f(ak)-f(ak-1)| \leqK

    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)|<= \sum ^{n}_{k=1}|f(ak)-f(ak-1)| \leqK

    but im not really sure how to prove the two statements
    any help is really appreciated
    thanks
    Just so our notation is straight. We have that V\left(\phi,P\right)=\sum_{j=1}^{n}\left|\phi\left  (x_j\right)-\phi\left(x_{j-1}\right)\right|. Then the variation of \phi on [a,b] is \sup_{P\in\mathcal{P}} V\left(\phi,P\right) where \mathcal{P} is the set of all partitions of [a,b].

    Then to prove that \phi is of bounded variation is simple since. We have that that \left|\phi(x)-\phi(k)\right|\leqslant \left|phi(x)-\phi(k)\right|+\left|\phi(x)-\phi(b)\right|\leqslant \text{Var}(\phi) and since \phi is of bounded variation we have that \text{Var}(\phi)<\infty. From where it follows that |\phi(x)|\leqslant \text{Var}(\phi)+|\phi(a)|<\infty, thus it is bounded.

    What do you have for the second one? Let me know.
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