Skip to main content
  • Textbook
  • © 2015

Real Analysis

Foundations and Functions of One Variable

  • 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)

Buying options

eBook USD 44.99
Price excludes VAT (USA)
  • ISBN: 978-1-4939-2766-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 59.99
Price excludes VAT (USA)
Hardcover Book USD 84.99
Price excludes VAT (USA)

This is a preview of subscription content, access via your institution.

Table of contents (20 chapters)

  1. Front Matter

    Pages i-x
  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

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 students' mathematical intuition.


The modular organization of the book makes it adaptable for either semester or year-long introductory courses, while the wealth of material allows for it to be used at various levels of student sophistication in all programs where analysis is a part of the curriculum, including teachers' education.

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. There are frequent hints and occasional complete solutions provided for the more challenging exercises making it an ideal choice for independent study.

The book includes a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a rigorous study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous historical notes and applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of mathematical analysis.

Keywords

  • Fourier series
  • Stieltjes integral
  • continuous functions
  • differentiation
  • infinite sequences
  • infinite series
  • integration
  • limits
  • real analysis

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: 978-1-4939-2765-4

  • Softcover ISBN: 978-1-4939-4222-0

  • eBook ISBN: 978-1-4939-2766-1

  • 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

Buying options

eBook USD 44.99
Price excludes VAT (USA)
  • ISBN: 978-1-4939-2766-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 59.99
Price excludes VAT (USA)
Hardcover Book USD 84.99
Price excludes VAT (USA)