The notions of upper sums and lower sums, which we analyzed for continuous functions, can in fact be

defined for any bounded function . However, for such functions part (iii) of the above theorem need not hold. One says a bounded is Riemann-integral or simply integral, if there exists a sequence of refinement partitions of and a real number such that

                            .

In fact, if for a function , there exists some sequence of refinement partitions such that

then for every sequences of refinement partitions, the upper and the lower sum sequences converge to the same limit, namely, .Thus, theorem 16.1.7 (iii) says the following:

 Every continuous function is integrable.

In fact, if is a bounded function such that has only finite number of discontinuities, say at then it can be shown that is integrable on and