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 Rn
Definition of the Derivative
Properties of the Differential
Further Properties of the Derivative
The Inverse Function Theorem
The Implicit Function Theorem
Higher Order Partial Derivatives
Higher 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
Biography
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.
"… intended for a first course in real analysis. … It could also be used to support an advanced calculus course. … The approach is theoretical and the writing rigorously mathematical. There are numerous exercises. … If a library needs to add to its collection in this area, this book would be a good choice. Summing up: Recommended. Upper-division undergraduates and graduate students."
—D. Z. Spicer, University System of Maryland, USA for CHOICE, October 2015"The book is carefully written, with rigorous proofs and a sufficient number of solved and unsolved problems. It is suitable for most university courses in mathematical analysis."
—Zentralblatt MATH 1317
