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 .