The Riemann Integral

Abstract

An account is given of the Riemann integral for real-valued functions defined on intervals of the real line, a rapid development of the topic made possible by use of the Darboux approach in place of that originally adopted by Riemann. The sense in which integration is the inverse of differentiation is investigated. To cope with the demands of the later chapters the improper Riemann integral is introduced. Uniform convergence of sequences and series is defined and its usefulness in interchanging integration and limits established; to help with circumstances in which uniform convergence is not present, Arzelà’s theorem is proved.