Real Analysis

About the Book

This is a user-friendly textbook for undergraduate real analysis (also called advanced calculus). It is self-contained, and with reasonable effort, the student should be able to read it without getting stuck. The emphasis is on using familiar examples to guide the reader to a thorough understanding of the abstract concepts of real analysis. It covers the standard topics of advanced calculus. The following is a brief outline of its contents:

Chapter 1 introduces elements of logic and methods of proving mathematical statements. Countable sets, uncountable sets and the Cantor set are also discussed.

Chapter 2 introduces, non-axiomatically, the usual topology on the real line and n-dimensional space. The main theorems one needs to discuss the various limit operations are then proved. Among them are the Bolzano-Weierstrass theorem, the Heine-Borel theorem and the classification of the connected subsets of the real line.

Chapter 3 introduces rigorous definitions of limits of sequences of real numbers. For many students, this is the chapter that introduces them to the "epsilon - N" arguments. Therefore, a number of examples and exercises involving "epsilon - N" arguments are included.

Chapter 4 addresses series of real numbers. Absolute and conditional convergence are defined and various tests for convergence are discussed. The proof of Riemann's theorem, that if a series converges conditionally then it is possible to rearrange it and get a series that converges to a pre-assigned real number, is given.

Chapter 5 starts with limits of functions of a real variable. The "epsilon - delta" arguments are discussed and the reader is given a number of examples and exercises to enable him/her master this important analytical tool. Limits are used to define the continuous, the uniformly continuous and the absolutely continuous functions of one variable. The basic properties of continuous functions defined on compact sets are proved and generalizations to functions of several variables are outlined.

Chapter 6 introduces a metric space and the topology induced by a metric. However, the theory developed here is not used extensively in the subsequent chapters. Therefore this chapter may be covered at any convenient time.

Chapter 7 discusses differentiable functions of one real variable. It starts with the idea that such a function f is differentiable at a point c in its domain if it is possible to draw a tangent to its graph at (c,f(c)). This leads to the view that f is differentiable at c if the difference f(c+h) - f(c) can be approximated to the first order by a map which is linear in h. The elementary properties of a derivative are derived, the mean value theorem is proved and a number of its consequences, like Taylor's theorem and the L'Hospital's rule are discussed. The inverse function theorem for a function of one variable is also proved.

Chapter 8 generalizes the results of Chapter 7 by discussing derivatives and higher order derivatives of a function of several variables. The inverse and implicit functions theorems for functions of several variables are proved, and several consequences of these theorems are discussed.

Chapter 9 introduces the Riemann and Darboux integrals of a bounded function f of one real variable and the two approaches to the integral are shown to be equivalent. The fundamental theorem of calculus for continuous functions is proved, the Stieltjes integral is defined and the variation of a function of one variable is discussed. Improper integrals and improper integrals depending on a parameter are also introduced and some of the consequences of uniform convergence of improper integrals depending on a parameter are discussed. The integrals of functions of several variables, including a proof of Fubini's theorem are also addressed.

Chapter 10 addresses sequences and series of functions. Pointwise and uniform convergence are defined and some consequences of uniform convergence are discussed. The Taylor and Fourier series are introduced, and sufficient conditions for the pointwise convergence of a Fourier series are discussed.

The last chapter, chapter 11, is intended to give the reader a view of the Lebesgue integral on the real line via a route that avoids a full-fledged measure theory course. An intuitive description of the sigma algebra generated by the open intervals is given, and the Lebesgue measure is introduced as a generalization of the length of an interval. The Lebesgue integral is defined as a limit of integrals of simple functions and the main convergence theorems are proved. The monotone convergence theorem is used to prove that the Lebesgue integral on the real line is an extension of the Riemann integral. It is also used to prove the change of variables theorem for multiple integrals.

About the Author

Dr. Frederick Semwogerere received his MS and PhD degrees in mathematics from Northwestern University and UC. Berkeley respectively. He has taught mathematics at Makerere University (Uganda), the University of Lesotho, UC. Berkeley (as a Graduate Student Instructor) and Clark Atlanta University, where he is currently an Associate Professor in the Mathematics Department