Real analysis: foundations and functions of one variable
Laczkovich, Miklos
- 5th ed.
- New York Springer 2015
- x, 483 p.
- Undergraduate Texts in Mathematics .
1. A Brief Historical Introduction 2. Basic Concepts 3. Laczkovich, Miklós (et al.) 4. Real Numbers 5. Infinite Sequences I 6. Infinite Sequences II 7. Infinite Sequences III 8. Rudiments of Infinite Series 9. Countable Sets 10. Real-Valued Functions of One Real Variable 11. Continuity and Limits of Functions 12. Various Important Classes of Functions (Elementary Functions) 13. Differentiation 14. Applications of Differentiation 15. The Definite Integral 16. Integration 17. Applications of Integration 18. Functions of Bounded Variation 19. The Stieltjes Integral 20. The Improper Integral 21. Erratum
Based on courses given at Eotvos Lorand University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable — systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student’s mathematical intuition. The book provides a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of the theoretical concepts treated. In the spirit of learning-by-doing, Real Analysis includes more than 500 engaging exercises for the student keen on mastering the basics of analysis. The wealth of material, and modular organization, of the book make it adaptable as a textbook for courses of various levels; the hints and solutions provided for the more challenging exercises make it ideal for independent study.