## Overview

- Revised and updated second edition with new material
- Text for a transition course between calculus and more advanced analysis courses
- Contains new material on topics such as irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions
- Includes new examples and improved proofs
- Includes supplementary material: sn.pub/extras

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

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## About this book

For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging.

The second edition preserves the book’s clear and concise style, illuminating discussions, and simple, well-motivated proofs. New topics include material on the irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions.

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## Keywords

- Bolzano-Weierstrass theorem
- L'Hospital's rule
- Riemann integral
- Riemann-Stieltjes integral
- Taylor's theorem
- continuous functions
- differentiation
- elementary analysis
- fundamental theorem of calculus
- integration
- limits of sequences
- mean value theorem
- monotone subsequences
- nowhere-differentiable functions
- power series
- rational zeros theorem

## Table of contents (7 chapters)

## Reviews

From the reviews of the first edition:

"This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis, such as continuity, convergence of sequences and series of numbers, and convergence of sequences and series of functions. There are many nontrivial examples and exercises, which illuminate and extend the material. The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and, in this reviewer’s opinion, has succeeded admirably."

—MATHEMATICAL REVIEWS

"This book occupies a niche between a calculus course and a full-blown real analysis course. … I think the book should be viewed as a text for a bridge or transition course that happens to be about analysis … . Lots of counterexamples. Most calculus books get the proof of the chain rule wrong, and Ross not only gives a correct proof but gives an example where the common mis-proof fails."

—Allen Stenger (The Mathematical Association of America, June, 2008)

## Authors and Affiliations

## About the author

Kenneth A. Ross is currently an emeritus professor of mathematics at the University of Oregon.

Jorge M. López is currently professor of mathematics at the University of Puerto Rico.

## Bibliographic Information

Book Title: Elementary Analysis

Book Subtitle: The Theory of Calculus

Authors: Kenneth A. Ross

Series Title: Undergraduate Texts in Mathematics

DOI: https://doi.org/10.1007/978-1-4614-6271-2

Publisher: Springer New York, NY

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

Copyright Information: Springer Science+Business Media, LLC, part of Springer Nature 2013

Hardcover ISBN: 978-1-4614-6270-5Published: 17 April 2013

Softcover ISBN: 978-1-4939-0128-9Published: 08 February 2015

eBook ISBN: 978-1-4614-6271-2Published: 16 April 2013

Series ISSN: 0172-6056

Series E-ISSN: 2197-5604

Edition Number: 2

Number of Pages: XII, 412

Topics: Analysis, Real Functions