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Numbers and Functions


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Table of Contents

Preface to first edition; Preface to second edition; Preface to third edition; Glossary; Part I. Numbers: 1 Mathematical induction; 2. Inequalities; 3. Sequences: a first bite at infinity; 4. Completeness: what the rational numbers lack; 5. Series: infinite sums; Part II. Functions: 6. Functions and continuity: neighbourhoods, limits of functions; 7. Continuity and completeness: functions on intervals; 8. Derivatives: tangents; 9. Differentiation and completeness: mean value theorems, Taylor's Theorem; 10. Integration: the fundamental theorem of calculus; 11. Indices and circle functions; 12. Sequences of functions; Appendix 1. Properties of the real numbers; Appendix 2. Geometry and intuition; Appendix 3. Questions for student investigation and discussion; Bibliography; Index.

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A revised and updated edition, providing hundreds of exercises to help students gradually transition from school to university-level calculus.

About the Author

R. P. Burn is an Honorary University Fellow at the University of Exeter. His other titles include Groups: A Path to Geometry (1985) and A Pathway into Number Theory (1982).


'This third edition of Numbers and Functions continues the author's long-term commitment to support every reader in making sense of mathematics by responding to a succession of well-chosen questions that encourage personal reflection and discussion with others. Groups of questions are followed by a summary to build the bigger picture. Every chapter includes details of the historical development and ends with a full list of solutions. The author is aware of the difficulties that students encounter with the complexity of the limit concept and begins with a pragmatic approach to null sequences. This broadens into a full study of limits of sequences, completeness, and a full range of tests for convergence of infinite series … This latest edition maintains the original chapters of the original, while benefiting from detailed improvements that have arisen from the experience of many readers. Thoroughly recommended.' David Tall, University of Warwick

'… written in a very comprehensible but exact way … an excellent guide through the basic course of mathematical analysis at university.' European Mathematical Society Newsletter

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