ABSTRACT
This chapter studies how many discontinuities a Riemann Integrable function can have and the composition of two Riemann Integrable functions. In general, the composition of Riemann Integrable functions is not Riemann integrable. The chapter develops the Riemann integral and computes lower, upper and Riemann sums for a given partition. The set of discontinuities of a monotone function is countable. Continuous functions with a finite number of discontinuities are integrable. The concept of length doesn’t seem to apply as there are no intervals in the discontinuity sets. The chapter describes Riemann — Lebesgue Lemma. This is also called Lebesgue’s Criterion for the Riemann Integrability of Bounded Functions. The chapter also reviews the equivalence classes of Riemann integrable functions.
