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Real Analysis

Foundations and Functions of One Variable

  • Textbook
  • © 2015

Overview

Authors:
  1. Miklós Laczkovich

    1. Department of Analysis, Eötvös Loránd University, Budapest, Hungary

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  2. Vera T. Sós

    1. Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, Budapest, Hungary

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    You can also search for this author in PubMed   Google Scholar

  • 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

Part of the book series: Undergraduate Texts in Mathematics (UTM)

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. A Brief Historical Introduction

    • Miklós Laczkovich, Vera T. Sós
    Pages 1-10

  3. Basic Concepts

    • Miklós Laczkovich, Vera T. Sós
    Pages 11-26

  4. Real Numbers

    • Miklós Laczkovich, Vera T. Sós
    Pages 27-50

  5. Infinite Sequences I

    • Miklós Laczkovich, Vera T. Sós
    Pages 51-62

  6. Infinite Sequences II

    • Miklós Laczkovich, Vera T. Sós
    Pages 63-75

  7. Infinite Sequences III

    • Miklós Laczkovich, Vera T. Sós
    Pages 77-87

  8. Rudiments of Infinite Series

    • Miklós Laczkovich, Vera T. Sós
    Pages 89-96

  9. Countable Sets

    • Miklós Laczkovich, Vera T. Sós
    Pages 97-102

  10. Real-Valued Functions of One Real Variable

    • Miklós Laczkovich, Vera T. Sós
    Pages 103-116

  11. Continuity and Limits of Functions

    • Miklós Laczkovich, Vera T. Sós
    Pages 117-166

  12. Various Important Classes of Functions (Elementary Functions)

    • Miklós Laczkovich, Vera T. Sós
    Pages 167-202

  13. Differentiation

    • Miklós Laczkovich, Vera T. Sós
    Pages 203-253

  14. Applications of Differentiation

    • Miklós Laczkovich, Vera T. Sós
    Pages 255-293

  15. The Definite Integral

    • Miklós Laczkovich, Vera T. Sós
    Pages 295-331

  16. Integration

    • Miklós Laczkovich, Vera T. Sós
    Pages 333-366

  17. Applications of Integration

    • Miklós Laczkovich, Vera T. Sós
    Pages 367-397

  18. Functions of Bounded Variation

    • Miklós Laczkovich, Vera T. Sós
    Pages 399-406

  19. The Stieltjes Integral

    • Miklós Laczkovich, Vera T. Sós
    Pages 407-416

  20. The Improper Integral

    • Miklós Laczkovich, Vera T. Sós
    Pages 417-437
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Keywords

About this book

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.

Reviews

“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 differential equations, upon which it touches occasionally. It can also serve as an introduction to Lebesgue integration or to generalized functions, which are not mentioned.” (Luis Manuel Braga de Costa Campos, Mathematical Reviews, December, 2016)

Authors and Affiliations

  • Department of Analysis, Eötvös Loránd University, Budapest, Hungary

    Miklós Laczkovich

  • Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Vera T. Sós

About the authors

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.

Bibliographic Information

  • Book Title: Real Analysis

  • Book Subtitle: Foundations and Functions of One Variable

  • Authors: Miklós Laczkovich, Vera T. Sós

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4939-2766-1

  • Publisher: Springer New York, NY

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer New York 2015

  • Hardcover ISBN: 978-1-4939-2765-4Published: 09 October 2015

  • Softcover ISBN: 978-1-4939-4222-0Published: 23 August 2016

  • eBook ISBN: 978-1-4939-2766-1Published: 08 October 2015

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: X, 483

  • Number of Illustrations: 94 b/w illustrations

  • Topics: Analysis

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