For any such partition, you can make a finer partition by adding c and d as "break points". Once you do that, the three sums are simple.
Suppose . Let be any non-zero real number. Define as follows:
(a) Prove that is Riemann Integrable on . What is its integral?
(b) If every in definition of were changed to and vice versa, how would the answer change?
For (a) the integral would be . Here is a proof. Proof. Let . Choose such that . Let be a partition of with . Choose arbitrarily such that . Then . We have to break this up into three sums? Because ultimately we want to show that .