A Course in Real Analysis

A Course in Real Analysis provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level. The book’s material has been extensively classroom tested in the author’s two-semester undergraduate course on real analysis at The George Washington University. The first part of the text presents the calculus of functions of one variable. This part covers traditional topics, such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions. It also includes optional sections on Stirling’s formula, functions of bounded variation, Riemann–Stieltjes integration, and other topics. The second part focuses on functions of several variables. It introduces the topological ideas (such as compact and connected sets) needed to describe analytical properties of multivariable functions. This part also discusses differentiability and integrability of multivariable functions and develops the theory of differential forms on surfaces in Rn. The third part consists of appendices on set theory and linear algebra as well as solutions to some of the exercises. A full solutions manual offers complete solutions to all exercises for qualifying instructors. With clear proofs, detailed examples, and numerous exercises, this textbook gives a thorough treatment of the subject. It progresses from single variable to multivariable functions, providing a logical development of material that will prepare students for more advanced analysis-based courses.

Functions of One Variable The Real Number System From Natural Numbers to Real Numbers Algebraic Properties of R Order Structure of R Completeness Property of R Mathematical Induction Euclidean Space Numerical Sequences Limits of Sequences Monotone Sequences Subsequences. Cauchy Sequences Limit Inferior and Limit Superior Limits and Continuity on R Limit of a Function Limits Inferior and Superior Continuous Functions Some Properties of Continuous Functions Uniform Continuity Differentiation on R Definition of Derivative. Examples The Mean Value Theorem Convex Functions Inverse Functions L’Hospital’s Rule Taylor’s Theorem on R Newton’s Method Riemann Integration on R The Riemann-Darboux Integral Properties of the Integral Evaluation of the Integral Stirling’s Formula Integral Mean Value Theorems Estimation of the Integral Improper Integrals A Deeper Look at Riemann Integrability Functions of Bounded Variation The Riemann-Stieltjes Integral Numerical Infinite Series Definition and Examples Series with Nonnegative Terms More Refined Convergence Tests Absolute and Conditional Convergence Double Sequences and Series Sequences and Series of Functions Convergence of Sequences of Functions Properties of the Limit Function Convergence of Series of Functions Power Series Functions of Several Variables Metric Spaces Definitions and Examples Open and Closed Sets Closure, Interior, and Boundary Limits and Continuity Compact Sets The Arzelà-Ascoli Theorem Connected Sets The Stone-Weierstrass Theorem Baire’s Theorem Differentiation on RnDefinition of the Derivative Properties of the Differential Further Properties of the Derivative The Inverse Function Theorem The Implicit Function Theorem Higher Order Partial DerivativesHigher Order Differentials. Taylor’s Theorem on Rn Optimization Lebesgue Measure on Rn Some General Measure Theory Lebesgue Outer Measure Lebesgue Measure Borel Sets Measurable Functions Lebesgue Integration on Rn Riemann Integration on Rn The Lebesgue Integral Convergence Theorems Connections with Riemann Integration Iterated Integrals Change of Variables Curves and Surfaces in Rn Parameterized Curves Integration on Curves Parameterized Surfaces m-Dimensional Surfaces Integration on Surfaces Differential Forms Integrals on Parameterized Surfaces Partitions of Unity Integration on m-Surfaces The Fundamental Theorems of Calculus Closed Forms in Rn Appendices A Set Theory B Summary of Linear Algebra C Solutions to Selected Problems Bibliography Index

Hugo D. Junghenn is a professor of mathematics at The George Washington University. He has published numerous journal articles and is the author of several books, including Option Valuation: A First Course in Financial Mathematics. His research interests include functional analysis, semigroups, and probability.