ABSTRACT
This chapter reviews the definition and some basic results on the theory of Riemann integration. It points out the limitations of Riemann integration so as to convince the reader of the necessity for a more general integral. Every bounded function f:[a,b]→R having at most a finite number of discontinuities is Riemann integrable and every monotonic function f:[a,b]→R is Riemann integrable. Thus, the set of all Riemann integrable functions is very large. An important property of R is its least upper bound property, stated as follows: Every subset of R which is bounded above has the least upper bound. Using the least upper bound property of R, it can be deduced that it has greatest lower bound property as well: Every subset of R which is bounded below has the greatest lower bound.
