ABSTRACT
In this chapter, the author explores certain aspects of Riemann Integration that are more subtle. He only proves the result for a closed finite interval. The general result for a compact subset of a more general set called a Topological Space is a modification of the proof which is actually not that more difficult, but that is another story. The author already knows that continuous functions, monotone functions and functions of bounded variation are classes of functions which are Riemann Integrable on the interval. He knows that the set of discontinuities of a monotone function is countable. The author shows that continuous functions with a finite number of discontinuities are integrable, and shows a function which was discontinuous on a countably infinite set and still was integrable! Hence, the author knows that a function is integrable should imply something about its discontinuity set.
