Problem Statement: If is bounded with finitely many discontinuities on then is Riemann Integrable on .
Proof: Let be the number of discontinuities of on and let be such that for every . We know such an exists since is bounded. Since is bounded we will be using the sup-inf definition of Riemann Integrable.
Denote a discontinuity by where . Let and . Let . By construction there are no discontinuities in any and so is Riemann Integrable one each .
Let . Then for each there exists a such that for any partition of with it follows that .
Let and let be a partition of with . Consider the sum below, we may separate the sum into parts, intervals which contain a discontinuity and those which do not.
Note that there are intervals and so there are terms less than . Also note that we have bounded each of the intervals that contain a discontinuity by , this is since the largest possible difference between the sup and inf is . Since there are discontinuities there are such terms in our inequality. Now we will use the fact that in order to simplify our inequality further.
Thus, we have shown that is Riemann Integrable on .