Preface to the Fourth Edition Preface to the Third Edition 1 Sets 1 Sets 2 Sets of real numbers 3 Countable and uncountable sets 4 Metric spaces 5 Open and closed sets 6 Dense and nowhere dense sets 7 Compactness 8 Convergence and completeness 9 Nested sets and Baire's theorem 10 Some applications of Baire's Theorem 11 Sets of measure zero 2 Functions 12 Functions 13 Continuous functions. 14 Properties of continuous functions 15 Upper and lower limits 16 Sequences of functions 17 Uniform convergence 18 Pointwise limits of continuous functions 19 Approximations to continuous functions 20 Linear functions 21 Derivatives 22 Monotonic functions 23 Convex functions 24 Infinitely differentiable functions 3 Integration 25 Lebesgue measure 26 Measurable functions 27 Definition of the Lebesgue integral 28 Properties of Lebesgue integrals 29 Applications of the Lebesgue integral 30 Stieltjes integrals 31 Applications of the Stieltjes integral 32 Partial sums of infinite series Answers to Exercises Index
