Undergraduate Texts in Mathematics

1 Undergraduate Texts in MathematicsEditorsS. RibetUndergraduate Texts in Mathematics (continued after index)Abbott:Understanding : Mathematics : A Concise History in :The Heritage of in :Introduction to Analytic NumberTheory. Second :Basic :Groups and :Linear Algebra Done Right. :Limits: A New Approach to :Complex Analysis. :Linear Algebra ThroughGeometry. Second :A First Course in Real :Conics and Cubics: A ConcreteIntroduction to Algebraic maud:An Introduction to :Factorization and :Second Year in :Mathematical Introduction toLinear Programming and Game :Mathematical Analysis: :Introduction to Rooij:Topological Spaces: FromDistance to :The Geometry of Spacetime: AnIntroduction to Special and Brunt:The Lebesgue StieltjesIntegral: A Practical :A Course in Modern :A Field Guide to AlgebraChilds:A Concrete Introduction to HigherAlgebra.

2 Second :Elementary ProbabilityTheory: With Stochastic Processes and anIntroduction to Mathematical Shea:Ideals, Varieties, andAlgorithms. Second :Basic Concepts of :Difference Rabbits to ChaosCurtis:Linear Algebra: An IntroductoryApproach. Fourth :Reading, Writing, andProving: A Closer Look at :The Joy of Sets: Fundamentalsof Contemporary Set Theory. Second :General :Why Math?Ebbinghaus/Flum/Thomas:Mathematical Logic. Second :Measure, Topology, and :An Introduction to DifferenceEquations. Third s/Sur nyi:Topics in the Theory :Practical Analysis on One :An Accompaniment to :Inside :The Fundamental Theoryof :Intermediate Real :Calculus Two: Linear andNonlinear Functions.

3 Second :Functions of Several :Combinatorial Optimization :Optimization Techniques: :Methods of :An Introduction to WaveletsThrough Linear :Complex :A Course in Calculus andReal AnalysisGordon:Discrete :Analysis by Its in :Finite-Dimensional Vector :Naive Set mmerlin/ in :Combinatorics andGraph :Geometry: Euclid and :Introduction to Calculus and :MathematicalReflections: In a Room with :Mathematical Vistas:From a Room with Many :Elementary Stability andBifurcation Theory. Second R. GhorpadeBalmohan V. LimayeA Course in Calculus and Real AnalysisWith 71 FiguresSudhir R. GhorpadeDepartment of MathematicsIndian Institute of Technology BombayPowai, Mumbai V.

4 LimayeDepartment of MathematicsIndian Institute of Technology BombayPowai, Mumbai Subject Classification (2000): 26-01 40-XXLibrary of Congress Control Number: 2006920312 ISBN-10: 0-387-30530-0 Printed on acid-free : 978-0387-30530-1 2006 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarlyanalysis. Use in connection with any form of information storage and retrieval, electronic adap-tation, computer software, or by similar or dissimilar methodology now known or hereafterdeveloped is use in this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether ornot they are subject to proprietary in the United States of America.

5 (MVY) Editors:S. AxlerMathematics DepartmentSan Francisco State UniversitySan Francisco, CA A. RibetDepartment of MathematicsUniversity of California at BerkeleyBerkeley, CA is one of the triumphs of the human mind. It emerged from inves-tigations into such basic questions as finding areas, lengths and volumes. Inthe third century , Archimedes determined the area under the arc of aparabola. In the early seventeenth century, Fermat and Descartes studied theproblem of finding tangents to curves. But the subject really came to life inthe hands of Newton and Leibniz in the late seventeenth century. In partic-ular, they showed that the geometric problems of finding the areas of planarregions and of finding the tangents to plane curves are intimately related toone another.

6 In subsequent decades, the subject developed further throughthe work of several mathematicians, most notably Euler, Cauchy, Riemann,and , calculus occupies a central place in Mathematics and is an essentialcomponent of Undergraduate education. It has an immense number of appli-cations both within and outside Mathematics . Judged by the sheer variety ofthe concepts and results it has generated, calculus can be rightly viewed as afountainhead of ideas and disciplines in analysis, often called mathematical analysis or simply analysis, maybe regarded as a formidable counterpart of calculus. It is a subject where onerevisits notions encountered in calculus, but with greater rigor and sometimeswith greater generality.

7 Nonetheless, the basic objects of study remain thesame, namely, real-valued functions of one or several real book attempts to give a self-contained and rigorous introduction tocalculus of functions of one variable. The presentation and sequencing of topicsemphasizes the structural development of calculus. At the same time, due im-portance is given to computational techniques and applications. In the courseof our exposition, we highlight the fact that calculus provides a firm foun-dation to several concepts and results that are generally encountered in highschool and accepted on faith. For instance, this book can help students get aclear understanding of (i) the definitions of the logarithmic, exponential andtrigonometric functions and a proof of the fact that these are not algebraicfunctions, (ii) the definition of an angle and (iii) the result that the ratio ofVIPrefacethe circumference of a circle to its diameter is the same for all circles.

8 It is ourexperience that a majority of students are unable to absorb these conceptsand results without getting into vicious circles. This may partly be due to thedivision of calculus and real analysis in compartmentalized courses. Calculusis often taught as a service course and as such there is little time to dwell onsubtleties and gain perspective. On the other hand, real analysis courses maystart at once with metric spaces and devote more time to pathological exam-ples than to consolidating students knowledge of calculus. A host of topicssuch as L H opital s rule, points of inflection, convergence criteria for Newton smethod, solids of revolution, and quadrature rules, which may have been in-adequately covered in calculus courses, become pass e when one studies realanalysis.

9 Trigonometric, exponential, and logarithmic functions are defined, ifat all, in terms of infinite series, thereby missing out on purely algebraic moti-vations for introducing these functions. The ubiquitous role of as a ratio ofvarious geometric quantities and as a constant that can be defined indepen-dently using calculus is often not well understood. A possible remedy wouldbe to avoid the separation of calculus and real analysis into seemingly disjointcourses and textbooks. Attempts along these lines have been made in the pastas in the excellent books of Hardy and of Courant and John. Ours is anotherattempt to give a unified exposition of calculus and real analysis and addressthe concerns expressed above.

10 While this book deals with functions of onevariable, we intend to treat functions of several variables in another genesis of this book lies in the notes we prepared for an undergraduatecourse at the Indian Institute of Technology Bombay in 1997. Encouraged bythe feedback from students and colleagues, the notes and problem sets wereput together in March 1998 into a bookletthat has been in private , it seemed that it would be relatively easy to convert that booklet intoa book. Seven years have passed since then and we now know a little better!While that booklet was certainly helpful, this book has evolved to acquire aform and philosophy of its own and is quite distinct from the original glance at the table of contents should give the reader an idea of thetopics covered.