ABSTRACT
This chapter proves that the set of Riemann integrable functions is quite rich and varied. To prove existence results for the Riemann integral, the Riemann sum approach is often awkward. Another way to approach it is to use Darboux sums including upper sums and lower sums. Upper and lower Darboux integrals and darboux integrability are described along with a new type of integrability for the bounded function. The chapter looks at the infimum of the upper sums and the supremum of the lower sums for a given bounded function. The two ways of defining integration are defined by a special criterion for integrability. The properties of the Riemann integral is reviewed along with fundamental infimum and supremum equalities. The Riemann integral exists on subintervals and is additive on subintervals.
