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Math Help - Integration by Darboux

  1. #1
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    Integration by Darboux

    Let h be a bounded function in [a,b] and let c be a point in [a,b]. Prove that for all e>0, there exists d>0 such that for every partition P of the [a,b] with parameter m(P)<d the following is true:
    |U(h,Q)-U(h,P)|<e
    where U(h,R) is the upper Darboux sum of h according to the partition R,
    and Q is the union of P and {c}.

    I think I can do most of the question myself, but I get stuck at the end. I'm not sure how to complete the proof for a case where c is a discontinuity of point of h.

    Thanks for help in advance.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Integration by Darboux

    Quote Originally Posted by durrrrrrrr View Post
    Let h be a bounded function in [a,b] and let c be a point in [a,b]. Prove that for all e>0, there exists d>0 such that for every partition P of the [a,b] with parameter m(P)<d the following is true:
    |U(h,Q)-U(h,P)|<e
    where U(h,R) is the upper Darboux sum of h according to the partition R,
    and Q is the union of P and {c}.

    I think I can do most of the question myself, but I get stuck at the end. I'm not sure how to complete the proof for a case where c is a discontinuity of point of h.

    Thanks for help in advance.
    Whether or not c is a discontinuity or not. The key is to make the partition P to be such that the points x,y for which c\in[x,y] are such that |x-y|\ll\varepsilon.
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  3. #3
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    Re: Integration by Darboux

    Ah. Thanks. I just got it.

    Is the following proof okay:
    U(f,Q) = area to the left of x + area between x and c + area between c and y + area to the right of y
    U(f,P) = area to the left of x + area between x and y + area to the right of y

    Can I then say that when y-x approaches 0 (refining the partition P more and more):
    U(f,P) = area to the left of x + area to the right of y = U(f,Q)

    Or is this too informal? How would I write this in a more formal manner?

    Thanks
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  4. #4
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    Re: Integration by Darboux

    How do I formalize the proof?
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