# A Course in Real Analysis

## Preview

## Book Description

**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 *R ^{n}*.

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.

## Table of Contents

*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 **R*n*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 R

*n*

Optimization

**Lebesgue Measure on **R*n *Some General Measure Theory

Lebesgue Outer Measure

Lebesgue Measure

Borel Sets

Measurable Functions

**Lebesgue Integration on **R*n *Riemann Integration on R

*n*

The Lebesgue Integral

Convergence Theorems

Connections with Riemann Integration

Iterated Integrals

Change of Variables

**Curves and Surfaces in **R*n *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 R

*n*

*Appendices *A Set Theory

B Summary of Linear Algebra

C Solutions to Selected Problems

Bibliography

Index

## Author(s)

### 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.

## Reviews

"… 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 forCHOICE, 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 MATH1317