Real Analysis

• 1. Introductory Ideas.- 1.1 Foreword for the Student: Is Analysis Necessary?.- 1.2 The Concept of Number.- 1.3 The Language of Set Theory.- 1.4 Real Numbers.- 1.5 Induction.- 1.6 Inequalities.- 2. Sequences and Series.- 2.1 Sequences.- 2.2 Sums, Products and Quotients.- 2.3 Monotonie Sequences.- 2.4 Cauchy Sequences.- 2.5 Series.- 2.6 The Comparison Test.- 2.7 Series of Positive and Negative Terms.- 3. Functions and Continuity.- 3.1 Functions, Graphs.- 3.2 Sums, Products, Compositions; Polynomial and Rational Functions.- 3.3 Circular Functions.- 3.4 Limits.- 3.5 Continuity.- 3.6 Uniform Continuity.- 3.7 Inverse Functions.- 4. Differentiation.- 4.1 The Derivative.- 4.2 The Mean Value Theorems.- 4.3 Inverse Functions.- 4.4 Higher Derivatives.- 4.5 Taylor’s Theorem.- 5. Integration.- 5.1 The Riemann Integral.- 5.2 Classes of Integrable Functions.- 5.3 Properties of Integrals.- 5.4 The Fundamental Theorem.- 5.5 Techniques of Integration.- 5.6 Improper Integrals of the First Kind.- 5.7 Improper Integrals of the Second Kind.- 6. The Logarithmic and Exponential Functions.- 6.1 A Function Defined by an Integral.- 6.2 The Inverse Function.- 6.3 Further Properties of the Exponential and Logarithmic Functions.- Sequences and Series of Functions.- 7.1 Uniform Convergence.- 7.2 Uniform Convergence of Series.- 7.3 Power Series.- 8. The Circular Functions.- 8.1 Definitions and Elementary Properties.- 8.2 Length.- 9. Miscellaneous Examples.- 9.1 Wallis’s Formula.- 9.2 Stirling’s Formula.- 9.3 A Continuous, Nowhere Differentiable Function.- Solutions to Exercises.- The Greek Alphabet.

From the reviews: Written in an easy-to-read style, combining informality with precision, the book is ideal for self-study or as a course textbook for first-and second-year undergraduates. Zentralblatt MATH ...the transition from the mysteries of real-analysis to the majesty of real analysis will be smoothed by this engaging, readable text. The Mathematical Gazette "This book is the distillation of Howie's considerable experience in teaching the introductory real analysis course: he adopts a concrete, pragmatic approach ... . The most striking feature of Real Analysis is ... the author's Ferrar-like concern for the reader's understanding which shines through on every page of his carefully written and carefully paced text. ... There are numerous worked examples and some 190 accessible, impeccably pitched exercises ... another attractive feature is the inclusion of full names and dates for all mathematicians mentioned." (Nick Lord, The Mathematical Gazette, Vol. 85 (504), 2001) "The book is a clear and structured introduction to real analysis. ... Fully worked out examples and exercises with solutions extend and illustrate the theory. Written in an easy-to-read style, combining informality and precision, the book is ideal for self-study or as a course textbook for first- and second-year undergraduates." (I. Rasa, Zentralblatt MATH, Vol. 969, 2001)