Introductory Analysis: The Theory of Calculus

Introductory Analysis, Second Edition, is intended for the standard course on calculus limit theories that is taken after a problem solving first course in calculus (most often by junior/senior mathematics majors). Topics studied include sequences, function limits, derivatives, integrals, series, metric spaces, and calculus in n-dimensional Euclidean space

Bases most of the various limit concepts on sequential limits, which is done firstDefines function limits by first developing the notion of continuity (with a sequential limit characterization)Contains a thorough development of the Riemann integral, improper integrals (including sections on the gamma function and the Laplace transform), and the Stieltjes integralPresents general metric space topology in juxtaposition with Euclidean spaces to ease the transition from the concrete setting to the abstract

New to This Edition

Contains new Exercises throughoutProvides a simple definition of subsequenceContains more information on function limits and L'Hospital's RuleProvides clearer proofs about rational numbers and the integrals of Riemann and StieltjesPresents an appendix lists all mathematicians named in the textGives a glossary of symbols

Introduction: Mathematical Statements and Proofs Ordering of the Real Numbers Sequence Limits Completeness of the Real Numbers Continuous Functions Consequences of Continuity The Derivative The Riemann Integral Improper Integrals Infinite Series The Riemann-Stieltjes Integral Function Sequences Power Series Metric Spaces and Euclidean Spaces Continuous Transformations Differential Calculus in Euclidean Spaces Area and Integration in E²

Appendix A. Mathematical Induction B. Countable and Uncountable Sets C. Infinite Products