分析入门 目录

Preface
1The Real Numbers
1.1Discussion: The Irrationality of 1.414
1.2Some Preliminaries
1.3The Axiom of Completeness
1.4Consequences of Completeness
1.5Cantor's Theorem
1.6Epilogue
2Sequences and Series
2.1Discussion: Rearrangements of Infinite Series
2.2The Limit of a Sequence
2.3The Algebraic and Order Limit Theorems
2.4The Monotone Convergence Theorem and a First Look at Infinite Series
2.5Subsequences and the Bolzano-Weierstrass Theorem
2.6The Cauchy Criterion
2.7Properties of Infinite Series
2.8Double Summations and Products of Infinite Series
2.9Epilogue
3 Basic Topology of R
3.1Discussion: The Cantor Set
3.2Open and Closed Sets
3.3Compact Sets
3.4Perfect Sets and Connected Sets
3.5Baire's Theorem
3.6Epilogue
4Functional Limits and Continuity
4.1Discussion: Examples of Dirichlet and Thomae
4.2Functional Limits
4.3Combinations of Continuous Functions
4.4Continuous Functions on Compact Sets
4.5The Intermediate Value Theorem
4.6Sets of Discontinuity
4.7Epilogue
5The Derivative
5.1Discussion: Are Derivatives Continuous?
5.2Derivatives and the Intermediate Value Property
5.3The Mean Value Theorem
5.4A Continuous Nowhere-Differentiable FunCtion
5.5Epilogue
6Sequences and Series of Functions
6.1Discussion: Branching Processes
6.2Uniform Convergence of a Sequence of Functions
6.3Uniform Convergence and Differentiation
6.4Series of Functions
6.5Power Series
6.6Taylor Series
6.7Epilogue
7The Riemann Integral
7.1Discussion: How Should Integration be Defined?
7.2The Definition of the Riemann Integral
7.3Integrating Functions with Discontinuities
7.4Properties of the Integral
7.5The Fundamental Theorem of Calculus
7.6Lebesgue's Criterion for Riemann Integrability
7.7Epilogue
8Additional Topics
8.1The Generalized Riemann Integral
8.2Metric Spaces and the Baire Category Theorem
8.3Fourier Series
8.4A Construction of R From Q
Bibliography
Index

分析入门 内容简介

My primary goal in writing Understanding Analysis was to create an elementary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it.There is a tendency, however, to center an introductory course too closely around the familiar theorems of the standard calculus sequence. Producing a rigorous argument that polynomials are continuous is good evidence for a well-chosen definition of continuity, but it is not the reason the subject was created and certainly not the reason it should be required study. By shifting the focus to topics where an untrained intuition is severely disadvantaged (e.g., rearrangements of infinite series, nowhere-differentiable continuous functions, Fourier series), my intent is to restore an intellectual liveliness to this course by offering the beginning student access to some truly significant achievements of the subject.