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An elementary text on the theory of functions of one real variable this book is intended for students with a good understanding of calculus as it begins with material on the real number system as a Dedekind complete ordered field and continuous functions. Pointwise and uniform convergence of series of functions, power series are discussed and a detailed treatment of differentiation, including differentiation of uniform limits of sequences is provided. Also included is an exhaustive treatment of both Riemann integration and Riemann-Stieltjes integration. Some features of this work, such as why Dedekind completeness is necessary for ensuring that nonconstant functions cannot have derivative zero everywhere without using multiple integration are exclusive to this book. There are several illustrative and worked examples, and a large number of problems with solutions (available on a compact disc).
Table of Contents
• Preface
• The Real Number System
• Continuous Functions
• Real and Complex Sequences
• Real and Complex Series
• Sequences and Series of Functions
• Uniform Approximation
• Differentiation
• More About Differentiation
• The Riemann Integral
• More About the Riemann Integral
• The Stieltjes Integral
• Hints
• References
• Index. |
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