If and if is Riemann Integrable on both and , then is Riemann Integrable on .
Proof: Since we know is Riemann Integrable on and it follows that is bounded on those intervals as well. Let be the maximum of those bounds, then it follows that for every .
Let . Since we know is Riemann Integrable on and , and bounded on , it follows that there exists a and such that for a partition of with and a partition of with it follows that where and where .
Let . Let be a partition of such that . Then is in at most two intervals of . This would happen if is an endpoint of an interval. Suppose and for some . Consider the sum below. We wish to show that the following sum is less than .
Thus, is Riemann Integrable on .
Reflection: This is a type of proof that shows up a lot on the quals. The trick is that we know we can make most of the sum small since we know it’s RI on a part of the interval. Since our function is bounded this let’s us pick a delta that works nicely. Then we can make everything “small” and so it falls out from there.