This book is the first of three volumes of a full and detailed account of those elements of real and complex analysis that mathematical undergraduates may expect to meet in the first two years or so of the study of analysis. This volume is concerned with the analysis of real-valued functions of a real variable. Volume II considers metric and topological spaces, and functions of several variables, while Volume III is concerned with complex analysis, and with the theory of measure and integration.

Mathematical analysis depends in a fundamental way on the properties of the real numbers, and indeed much of analysis consists of working out their consequences. It is therefore essential to develop a full understanding of these properties. There are two ways of doing this. The traditional and appropriate way is to take the fundamental properties of the real numbers as axioms – the real numbers form an ordered field in which every non-empty subset which has an upper bound has a least upper bound – and to develop the theory – convergence, continuity, differentiation and integration – from these axioms. This programme is carried out in Part Two. This theory is meant to be used, and Part Two ends with an extensive collection of applications. The reader is strongly recommended to follow this tradition, and to begin at the beginning of Part Two.