1. Sequences.- 1.1 Examples, Formulae and Recursion.- 1.2 Monotone and Bounded Sequences.- 1.3 Convergence.- 1.4 Subsequences.- 1.5 Cauchy Sequences.- Exercises.- 2. Functions and Continuity.- 2.1 Examples.- 2.2 Monotone and Bounded Functions.- 2.3 Limits and Continuity.- 2.4 Bounds and Intermediate Values.- 2.5 Inverse Functions.- 2.6 Recursive Limits and Iteration.- 2.7 One-Sided and Infinite Limits. Regulated Functions.- 2.8 Countability.- Exercises.- 3. Differentiation.- 3.1 Differentiable Functions.- 3.2 The Significance of the Derivative.- 3.3 Rules for Differentiation.- 3.4 Mean Value Theorems and Estimation.- 3.5 More on Iteration.- 3.6 Optimisation.- Exercises.- 4. Constructive Integration.- 4.1 Step Functions.- 4.2 The Integral of a Regulated function.- 4.3 Integration and Differentiation.- 4.4 Applications.- 4.5 Further Mean Value Theorems.- Exercises.- 5. Improper Integrals.- 5.1 Improper Integrals on an Interval.- 5.2 Improper Integrals at Infinity.- 5.3 The Gamma function.- Exercises.- 6. Series.- 6.1 Convergence.- 6.2 Series with Positive Terms.- 6.3 Series with Arbitrary Terms.- 6.4 Power Series.- 6.5 Exponential and Trigonometric Functions.- 6.6 Sequences and Series of Functions.- 6.7 Infinite Products.- Exercises.- 7. Applications.- 7.1 Fourier Series.- 7.2 Fourier Integrals.- 7.3 Distributions.- 7.4 Asymptotics.- 7.5 Exercises.- A. Fubini's Theorem.- B. Hints and Solutions for Exercises.

This book adopts a practical, example-led approach to mathematical analysis that shows both the usefulness and limitations of the results. A number of applications show what the subject is about and what can be done with it; the applications in Fourier theory, distributions and asymptotics show how the results may be put to use. Exercises at the end of each chapter, of varying levels of difficulty, develop new ideas and present open problems.