Theorem. Let . Let be the smallest closed interval containing and and suppose that . If is a function, then the integral exists if and only if the integrals and exist. So .
Outline of Proof. (1) We want to show that is Riemann integrable on and . Let . Try to find a corresponding . Let and be partitions of and respectively with meshes less than . Then look at . (2) Show that any Riemann sum of on is within of . We know that is bounded on . Suppose . Find a corresponding to on and corresponding to on . Choose where is a function of and . Now consider the following:
Here we have to be careful about the endpoints?
Is this in general correct? Any other faster ways of proving this?