© 2018

A Course in Calculus and Real Analysis


  • Offers a unified exposition of single-variable calculus and classical real analysis

  • Contains a chapter on sequences and series of real-valued functions of a real variable

  • Features two new appendices that offer a construction of real numbers


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

Table of contents

  1. Front Matter
    Pages I-IX
  2. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 1-40
  3. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 41-66
  4. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 67-104
  5. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 105-148
  6. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 149-180
  7. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 181-232
  8. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 233-294
  9. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 295-364
  10. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 365-424
  11. Sudhir R. Ghorpade, Balmohan V. Limaye
    Pages 425-502
  12. Back Matter
    Pages 503-538

About this book


Offering a unified exposition of calculus and classical real analysis, this textbook presents a meticulous introduction to single‐variable calculus. Throughout, the exposition makes a distinction between the intrinsic geometric definition of a notion and its analytic characterization, establishing firm foundations for topics often encountered earlier without proof. Each chapter contains numerous examples and a large selection of exercises, as well as a “Notes and Comments” section, which highlights distinctive features of the exposition and provides additional references to relevant literature.

This second edition contains substantial revisions and additions, including several simplified proofs, new sections, and new and revised figures and exercises. A new chapter discusses sequences and series of real‐valued functions of a real variable, and their continuous counterpart: improper integrals depending on a parameter. Two new appendices cover a construction of the real numbers using Cauchy sequences, and a self‐contained proof of the Fundamental Theorem of Algebra.

In addition to the usual prerequisites for a first course in single‐variable calculus, the reader should possess some mathematical maturity and an ability to understand and appreciate proofs. This textbook can be used for a rigorous undergraduate course in calculus, or as a supplement to a later course in real analysis. The authors’ A Course in Multivariable Calculus is an ideal companion volume, offering a natural extension of the approach developed here to the multivariable setting.

From reviews:
[The first edition is] a rigorous, well-presented and original introduction to the core of undergraduate mathematics — first-year calculus. It develops this subject carefully from a foundation of high-school algebra, with interesting improvements and insights rarely found in other books. […] This book is a tour de force, and a necessary addition to the library of anyone involved in teaching calculus, or studying it seriously.  N.J. Wildberger, Aust. Math. Soc. Gaz.


Honors Calculus textbook Introduction to Real Analysis Ghorpade Limaye textbook real analysis textbook Real Functions Sequences and Series Continuity and Limits Differentiation Integration transcendental functions Riemann integrals Infinite Series

Authors and affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayPowai, Mumbai 400076India
  2. 2.Department of MathematicsIndian Institute of Technology DharwadDharwad, Karnataka 580011India

About the authors

Sudhir R. Ghorpade is Institute Chair Professor in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He has received several awards, including the All India Council for Technical Education (AICTE) Career Award for Young Teachers and the Prof. S.C. Bhattacharya Award for Excellence in Pure Sciences. His research interests lie in algebraic geometry, combinatorics, coding theory, and commutative algebra.

Balmohan V. Limaye is Professor Emeritus in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He is the author of several research monographs and textbooks, including Linear Functional Analysis for Scientists and Engineers (Springer, 2016). He worked at IIT Bombay for more than 40 years and has twice received the Award for Excellence in Teaching from IIT Bombay. His research interests include Banach algebras, approximation theory, numerical functional analysis, and linear algebra.

The authors’ companion volume A Course in Multivariable Calculus and Analysis (2010) is also in the UTM series.

Bibliographic information

  • Book Title A Course in Calculus and Real Analysis
  • Authors Sudhir R. Ghorpade
    Balmohan V. Limaye
  • Series Title Undergraduate Texts in Mathematics
  • Series Abbreviated Title Undergraduate Texts Mathematics
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-01399-8
  • Softcover ISBN 978-3-030-82741-0
  • eBook ISBN 978-3-030-01400-1
  • Series ISSN 0172-6056
  • Series E-ISSN 2197-5604
  • Edition Number 2
  • Number of Pages IX, 538
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Calculus
    Real Functions
    Sequences, Series, Summability
  • Buy this book on publisher's site


“This book would be a valuable asset to a university library and that many instructors would do well to have a copy of this book in their personal libraries. In addition, I believe that some students would benefit if they possessed a copy of this book to use for reference purposes.” (Jonathan Lewin, MAA Reviews, April 15, 2019)