1. ## 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.

2. ## Re: Integration by Darboux

Originally Posted by durrrrrrrr
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.

Whether or not $\displaystyle c$ is a discontinuity or not. The key is to make the partition $\displaystyle P$ to be such that the points $\displaystyle x,y$ for which $\displaystyle c\in[x,y]$ are such that $\displaystyle |x-y|\ll\varepsilon$.

3. ## 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

4. ## Re: Integration by Darboux

How do I formalize the proof?