Based on courses given at E?tv?s Lor?nd 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 students 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.
A Short Historical Introduction.- Basic Concepts.- Real Numbers.- Infinite Sequences I.- Infinite Sequences II.- Infinite Sequences III.- Rudiments of Infinite Series.- Countable Sets.- Real Valued Functions of One Variable.- Continuity and Limits of Functions.- Various Important Classes of Functions (Elementary Functions).- Differentiation.- Applications of Differentiation.- The Definite Integral.- Integration.- Applications of Integration.- Functions of Bounded Variation.- The Stieltjes Integral.- The Improper Integral.
This book is written to be accessible to the competent university student. & The book is consistent in addressing the classical analysis of real functions of one real variable, and it can serve as an introduction to monographs of complex functions, functional analysis and dilól
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