Numbers and Functions: Steps into Analysis

Citation

Burn, R. P. (2015). Numbers and Functions: Steps into Analysis (3rd ed.). Cambridge University Press.

Chapter Summary

The book is a sequence of more than 800 numbered questions, most with brief answers, arranged in twelve chapters in two parts, "Numbers" and "Functions", with three short appendices. The reader works the questions rather than reading proofs. Chapters open with suggested preliminary, concurrent and further reading (Chapter 1, for instance, points to Pólya and to Sominskii on induction). Summaries appear whenever a major idea has been rounded off, and historical notes and answers follow. The first edition appeared in 1992 and the second in 2000; the third edition (2015) corrects details and reworks a few questions in Chapters 3 and 10.

Chapter 1: Mathematical induction

The opening chapter builds the habit of proof by induction. Its first questions test a formula for small cases, then show how truth for one value carries to the next. According to the first-edition preface, Chapters 1 and 2 are meant for students who need to strengthen technique and confidence before the analysis proper begins.

Chapter 2: Inequalities

Most questions here derive the basic properties of the number system and of inequalities from stated axioms. The preface presents this as practice in telling apart what is familiar from what has actually been proved.

Chapter 3: Sequences: a first bite at infinity

Limits are introduced through null sequences. In the third edition this introduction builds on Archimedes' method for finding areas and volumes. The chapter develops convergence of sequences without yet assuming completeness.

Chapter 4: Completeness: what the rational numbers lack

The book takes "every infinite decimal is convergent" as its axiom of completeness and derives from it the convergence of bounded monotonic sequences. The preface admits that this takes more space than the usual route and defends it as keeping the student's understanding active. Least upper bounds were given a larger role in the second edition.

Chapter 5: Series: infinite sums

The last chapter of Part I turns to infinite sums. David Tall's endorsement for the third edition describes how the book moves from null sequences to limits of sequences, completeness and a full range of tests for the convergence of infinite series.

Chapter 6: Functions and continuity: neighbourhoods, limits of functions

Part II opens with functions, neighbourhoods and limits of functions. The preface explains that this chapter builds on Chapter 3 but does not depend on completeness. It also explains the author's preference for defining continuity through convergent sequences, which he found more effective in teaching than the neighbourhood definition.

Chapter 7: Continuity and completeness: functions on intervals

This chapter proves the harder theorems about continuous functions on intervals, all of which depend on completeness. The proofs use convergent subsequences of bounded sequences.

Chapter 8: Derivatives: tangents

The definition of the derivative and the product, quotient and chain rules, the properties that do not depend on completeness. The differentiation of inverse functions is also placed here.

Chapter 9: Differentiation and completeness: mean value theorems, Taylor's Theorem

The results that do need completeness, from Rolle's Theorem through the mean value theorems to Taylor's Theorem.

Chapter 10: Integration: the fundamental theorem of calculus

The chapter starts by computing areas in ways that were, or could have been, used before Newton. The third edition's opening questions find areas under continuous monotonic curves without appealing to completeness. It then uses step functions to build the Riemann integral, with completeness entering through upper and lower integrals, and ends with the Fundamental Theorem of Calculus.

Chapter 11: Indices and circle functions

Instead of defining the exponential, logarithm and trigonometric functions by integrals or power series, the book starts from indices (powers) and from the length of an arc of a circle. The preface concedes that this is longer than usual but notes that it applies the theorems of the first ten chapters.

Chapter 12: Sequences of functions

A closing chapter on uniform convergence, which some courses leave to the second year. It discusses term-by-term integration and differentiation, completing a university-style treatment of school calculus.

Appendices 1–3: Properties of the real numbers; Geometry and intuition; Questions for student investigation and discussion

Short appendices list the properties of the real numbers that the course relies on, comment on geometry and intuition, and offer open questions for investigation and discussion.

Key Concepts

Chapters 1–2: Mathematical induction; Inequalities

• Proof by induction: showing that truth for one natural number carries to the next, after testing small cases.
• Axiomatic properties of numbers: deriving familiar facts about order and inequalities from stated axioms instead of taking them for granted.

Chapters 3–5: Sequences; Completeness; Series

• Null sequences: sequences tending to zero, the book's starting point for limits.
• Completeness via infinite decimals: the axiom that every infinite decimal converges, from which bounded monotonic sequences are shown to converge.
• Series as limits: an infinite sum is the limit of its partial sums.

Chapters 6–7: Continuity

• Sequential continuity: continuity defined through convergent sequences, the book's preferred definition.
• Continuity on intervals: theorems that depend on completeness, proved through convergent subsequences of bounded sequences.

Chapters 8–10: Derivatives and integration

• Derivative rules without completeness: product, quotient and chain rules follow from the definition alone.
• Mean value theorems and Taylor's Theorem: results that require completeness.
• Riemann integral via step functions: upper and lower integrals built from step functions.

Chapters 11–12: Indices and circle functions; Sequences of functions

• Elementary functions from first principles: exponentials and logarithms developed from indices, circular functions from arc length.
• Uniform convergence: the condition that justifies term-by-term integration and differentiation.

Critical Analysis

Strengths

• Learning by constructing. The first-edition preface sets out the principles behind the design: let students use what they already know to generate new concepts, and move from carefully chosen special cases to the general theorem. The questions are built so that readers formulate definitions and theorems themselves.
• Rigour with attention to its foundations. The author says the book insists, more than is usual, that students know which parts of the axiomatic basis are needed at each point. The chapter titles reflect this: continuity, differentiation and, since the third edition, integration are each split into what can be done without completeness and what needs it.
• Tested and endorsed by teachers of analysis. The second-edition preface records that many problems from the first five chapters were used in the problem-solving approach to first-year analysis adopted at the University of Warwick. That experience led to more summaries and diagrams and short names for small theorems. David Tall's endorsement for the third edition praises the well-chosen succession of questions and the summaries that build the larger picture. The European Mathematical Society Newsletter called the book comprehensible but exact.
• History as a guide. Historical notes run through the book. The preface argues that school calculus works at roughly eighteenth-century standards of rigour, so retracing the nineteenth-century development is a natural route to modern analysis.

Limitations

• It cannot be skimmed. With more than 800 questions and most of the theory carried by them, the book gives little to a reader who does not work the problems.
• Some choices are non-standard. Completeness as convergence of infinite decimals, continuity defined through sequences, and exponential and circular functions built from indices and arc length differ from the definitions most lecture courses use. The author acknowledges that some of these routes take longer. Students will need to connect them with their own course's definitions.
• Assumptions of the author's era. The first-edition preface assumes a scientific calculator and access to graph-drawing software. It also assumes a British sixth-form calculus background, which readers from other systems may need to adapt to.

Where It Fits in the Course

Of the books in this course, Burn's is the one built for doing. Lara Alcock's How to Think About Analysis is a study guide that explains how to read definitions and proofs. Burn instead makes the student construct them through problems. The two pair well before or during a first analysis course: Alcock for the thinking habits, Burn for sustained practice on the same topics.

As the main text for a first rigorous course, Stephen Abbott's Understanding Analysis presents the theory in continuous prose with complete proofs, and many students will use Burn's questions to prepare for or reinforce its early chapters. Bartle and Sherbert's Introduction to Real Analysis is a more comprehensive textbook. Walter Rudin's Principles of Mathematical Analysis is the formal classic for later study. Victor Bryant's Yet Another Introduction to Analysis is another introductory treatment of the same core material. The author presents the book as a springboard to metric spaces, point-set topology or multivariable calculus.

How to Use This Book

The book is aimed at students moving from school calculus to university analysis: in the author's words, readers who have studied calculus at school and are ready to review it rigorously. School algebra and calculus are the prerequisites, together with a willingness to work with pen and paper.

• Work every question, in order. The theory lives in the questions. Attempt each before checking the brief answer, and use the summaries to consolidate.
• Follow the dependency map. The preface gives a chart of how the chapters depend on each other. Chapters 1–2 can be skipped by students already confident with induction and inequalities. Chapters 3, 4 and 5 form the core on sequences, completeness and series. Chapter 6 needs only Chapter 3, and Chapters 7, 9 and 10 draw on completeness.
• Match it to your course. For a sequences-and-series course, focus on Chapters 3–5. For continuity, differentiation and integration, work Chapters 6–10. Chapter 11 on elementary functions and Chapter 12 on uniform convergence are natural extensions, and the preface notes that some courses defer uniform convergence to the second year.
• Use it in a group. The questions suit workshops and paired work, and Appendix 3 provides questions for investigation and discussion.