Numbers and Functions
Steps into Analysis

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

R. P. Burn (Author)

9781107444539, Cambridge University Press

Paperback / softback, published 19 February 2015

374 pages, 65 b/w illus. 800 exercises
22.8 x 15.1 x 2.1 cm, 0.55 kg

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

The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this third edition of Numbers and Functions, Professor Burn invites the student reader to tackle each of the key concepts in turn, progressing from experience through a structured sequence of more than 800 problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, of which most are supplied with brief answers, draws students into constructing definitions and theorems for themselves. This natural development is informed and complemented by historical insight. Carefully corrected and updated throughout, this new edition also includes extra questions on integration and an introduction to convergence. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties.

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.

Subject Areas: Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB], Mathematics [PB]