Synopses & Reviews
This is a revised, updated, and augmented edition of a classic Carus monograph with a new chapter on integration and its applications. Earlier editions covered sets, metric spaces, continuous functions, and differentiable functions. To that, this edition adds sections on measurable sets and functions and the Lebesgue and Stieltjes integrals. The book retains the informal chatty style of the previous editions. It presents a variety of interesting topics, many of which are not commonly encountered in undergraduate textbooks, such as the existence of continuous everywhere-oscillating functions; two functions having equal derivatives, yet not differing by a constant; application of Stieltjes integration to the speed of convergence of infinite series. For readers with a background in calculus, the book is suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Students of mathematics will find here the sense of wonder that was associated with the subject in its early days.
Revised edition of a classic Carus monograph with a new chapter on integration and its applications.
Table of Contents
Part I. 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 problem; 10. Some applications of Baire's theorem; 11. Sets of measure zero; Part II. 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; Part III. 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.