Let for and . Show that is Riemann integrable on .

I have a feeling that I have to use some fact like is monotone on some intervals but I really don't know. Any help would be appreciated.

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- May 10th 2010, 07:04 PMPinkkProve integrability
Let for and . Show that is Riemann integrable on .

I have a feeling that I have to use some fact like is monotone on some intervals but I really don't know. Any help would be appreciated. - May 10th 2010, 07:11 PMJose27
I'ts easy to prove that if a function is discontinous only at a finite number of points (and bounded of course) then it is Riemann integrable (use induction and isolate the point of discontinuity). Or more generally Lebesgue's criterion for integrability which says a function is Riemann integrable iff the set on which said function is discontinous has measure 0.

- May 10th 2010, 07:20 PMPinkk
Well, the point of discontinuity is , but could you show how only a finite number of discontinuities and boundedness implies integrability?

- May 10th 2010, 07:22 PMDrexel28
- May 10th 2010, 07:37 PMPinkk
I don't see how the inf of the function on that interval would be 0 since it can take on negative values on such an interval. But I guess the general idea would be that and the same for the lower sums, correct? (I know that is technically an incorrect statement since we have to state that the union of the two partitions is in fact the partition of the larger interval and that a is an element in both partitions, etc, etc).

- May 10th 2010, 07:41 PMJose27
It's easy: Given just divide into where and for all then in the first and last intervals the function is continous and in the second just use an argument analogous to Drexel's (if the discontinuity is on one of the endpoints the argument is the same). The result follows by induction.

- May 10th 2010, 07:42 PMDrexel28