Real Analysis

Based on courses given at Eötvös Loránd 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.

Includes insightful historical remarks regarding real analysis Presents core ideas of analysis as a way of thinking as opposed to a body of facts Explains how and why ideas arise, then how they evolve to the mature notions of real analysis

Auteur
Miklós Laczkovich is Professor of Mathematics at Eötvös Loránd University and the University College London, and was awarded the Ostrowski Prize in 1993 and the Széchenyi Prize in 1998. Vera T. Sós is a Research Fellow at the Alfréd Rényi Institute of Mathematics, and was awarded the Széchenyi Prize in 1997.

Contenu

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.