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Math Help - Proof involving Lipschitz continuity with constant L

  1. #1
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    Cool Proof involving Lipschitz continuity with constant L

    Problem:
    Prove that if the function f:[a,b]-->R is Lipschitz continuous with constant L then for every partition P of [a,b]
    U(f,P)-L(f,P) <= ||P||*L*(b-a)




    Okay so from Lipschitz continuity I see that dY(f(a),f(b)) <= L*dx(a,b) for all a,b in X

    So dx(a,b) is the same as (b-a)
    Then take a partition P with ||P|| < delta for both sides

    Where do i go from here .... i nearly have the right side completed .... or so i think but what should my next move be when it comes to the left side? How does dY(f(a),f(b)) become U(f,P)-L(f,P)?

    PLease help
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by derek walcott View Post
    Problem:
    Prove that if the function f:[a,b]-->R is Lipschitz continuous with constant L then for every partition P of [a,b]
    U(f,P)-L(f,P) <= ||P||*L*(b-a)




    Okay so from Lipschitz continuity I see that dY(f(a),f(b)) <= L*dx(a,b) for all a,b in X

    So dx(a,b) is the same as (b-a)
    Then take a partition P with ||P|| < delta for both sides

    Where do i go from here .... i nearly have the right side completed .... or so i think but what should my next move be when it comes to the left side? How does dY(f(a),f(b)) become U(f,P)-L(f,P)?

    PLease help
    Why are you using the metric notation?

    U(f,P)-L(f,P)=\left|U(f,p)-L(f,p)\right| =\left|\sum_{j=1}^{n}(M_j-m_j)\Delta x_j\right|\leqslant\sum_{j=1}^{n}|M_j-m_j|\Delta x_j. Now, M_j=f(x*) for some x*\in[x_{j-1},x_j] and similarly for m_j. But, by Lipschitz |M_j-m_j|=|f(x*)-f(y*)|\leqslant L|x_{j}-x_{j-1}|\leqslant L\|P\|. Thus, \left|U(P,f)-L(P,f)\right|\leqslant\sum_{j=1}^{n}L\|P\|\Delta x_j=L\|P\|\sum_{j=1}^{n}\Delta x_j=L\|P\|(b-a).
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  3. #3
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    oh so i shouldn't start with dY(f(a), f(b)) and instead start with U(f,P)-L(f,P) and work towards L*||P||*(b-a)

    what exactly does metric notation mean and how does that differ from the notation you are using?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by derek walcott View Post
    oh so i shouldn't start with dY(f(a), f(b)) and instead start with U(f,P)-L(f,P) and work towards L*||P||*(b-a)

    what exactly does metric notation mean and how does that differ from the notation you are using?
    In this context it doesn't really matter. But,the real numbers have much more than a topological structure (a metric) they have an algebraic structure (it is a field). Consequently while the true definition of a Lipschitz continuous function from a metric space X to a metric space Y is d_Y(f(x),f(y))\leqslant \delta d_x(x,y) writing it like that when you have the idea of addition and subtraction can do nothing but confuse you.
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