This book introduces the elements of the theory of real-valued functions of a real variable. The book is aimed at young mathematicians and others who would like to see a coherent account of basic analysis as a rigorous mathematical theory. It aims to avoid various extremes: It does not brush any of the serious difficulties in analysis proper under the carpet. The reader is assumed to be bright, and willing to work hard. It does not dwell on routine stuff. The reader has probably already taken a non-rigorous calculus course, and does not need to be led through that material again.

This book leads up to the starting-point of a rigorous course in basic real analysis. It is designed to satisfy a student who wants to start 'further back' than the axioms of a complete ordered field. Typically, such a student will be in the second or higher year at University, and will have attained some level of mathematical maturity. The book presents a foundation in set theory, and builds up through the natural numbers, integers, and rational numbers to the real and complex numbers, and establishes their properties on the basis of some more basic axioms.

Reversibility is a thread woven through many branches of mathematics. It arises in dynamics, in systems that admit a time-reversal symmetry, and in group theory where the reversible group elements are those that are conjugate to their inverses. However, the lack of a lingua franca for discussing reversibility means that researchers who encounter the concept may be unaware of related work in other fields. This text is the first to make reversibility the focus of attention. The authors fix standard notation and terminology, establish the basic common principles, and illustrate the impact of reversibility in such diverse areas as group theory, differential and analytic geometry, number theory, complex analysis and approximation theory. As well as showing connections between different fields, the authors' viewpoint reveals many open questions, making this book ideal for graduate students and researchers. The exposition is accessible to readers at the advanced undergraduate level and above.

This book leads up to the starting-point of a rigorous course in basic real analysis. It is designed to satisfy a student who wants to start 'further back' than the axioms of a complete ordered field. Typically, such a student will be in the second or higher year at University, and will have attained some level of mathematical maturity. The book presents a foundation in set theory, and builds up through the natural numbers, integers, and rational numbers to the real and complex numbers, and establishes their properties on the basis of some more basic axioms.