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Introduction to Real Analysis: An Educational Approach

ISBN: 978-1-118-16441-9

September 2011

280 pages

Description
An accessible introduction to real analysis and its connection to elementary calculus

Bridging the gap between the development and history of real analysis, Introduction to Real Analysis: An Educational Approach presents a comprehensive introduction to real analysis while also offering a survey of the field. With its balance of historical background, key calculus methods, and hands-on applications, this book provides readers with a solid foundation and fundamental understanding of real analysis.

The book begins with an outline of basic calculus, including a close examination of problems illustrating links and potential difficulties. Next, a fluid introduction to real analysis is presented, guiding readers through the basic topology of real numbers, limits, integration, and a series of functions in natural progression. The book moves on to analysis with more rigorous investigations, and the topology of the line is presented along with a discussion of limits and continuity that includes unusual examples in order to direct readers' thinking beyond intuitive reasoning and on to more complex understanding. The dichotomy of pointwise and uniform convergence is then addressed and is followed by differentiation and integration. Riemann-Stieltjes integrals and the Lebesgue measure are also introduced to broaden the presented perspective. The book concludes with a collection of advanced topics that are connected to elementary calculus, such as modeling with logistic functions, numerical quadrature, Fourier series, and special functions.

Detailed appendices outline key definitions and theorems in elementary calculus and also present additional proofs, projects, and sets in real analysis. Each chapter references historical sources on real analysis while also providing proof-oriented exercises and examples that facilitate the development of computational skills. In addition, an extensive bibliography provides additional resources on the topic.

Introduction to Real Analysis: An Educational Approach is an ideal book for upper- undergraduate and graduate-level real analysis courses in the areas of mathematics and education. It is also a valuable reference for educators in the field of applied mathematics.

About the Author
WILLIAM C. BAULDRY, PhD, is Professor in the Department of Mathematical Sciences at Appalachian State University. Dr. Bauldry has published numerous articles in his areas of interest, which include the pedagogical uses of computer algebra systems and cryptography.
Features
  • Provides a unique organization of the topic; a brief review of basic concepts of calculus is followed by an introduction to real analysis and a thorough discussion of measure theory
  • Contains numerous exercises and examples that are both proof oriented and facilitate an understanding of computational skills 
  • Engages readers by beginning with an AP calculus focus, and then quickly moving to the more theoretical aspects of mathematical analysis topics
  • Thoroughly classroom-tested material based on the author's own Real Analysis for Teachers course
  • Is accompanied by supplemental material including a bibliography of calculus, analysis, and pedagogical references as well as appendices that contain the main theorems of real analysis and a collection of student projects to be used in the classroom