內容簡介
這《分析入門》在很大程度上更好的挑戰和提高數學直覺而非改變它，其本質是揭示分析的內在魅力所在。全書將Cantor集是否包括所有的無理數、函式不連續點的集合是否可以是任意集合、導數是否連續、導函式是否可積以及一個無窮次可微函式是否一定是一個泰勒級數的極限這些問題作為《分析入門》的一條主線，使得《分析入門》脈絡清晰、易懂。
目錄
Preface
1 The Real Numbers
1.1 Discussion: The Irrationality of vz2
1.2 Some Preliminaries
1.3 The Axiom of Completeness
1.4 Consequences of Completeness
1.5 Cantor's Theorem
1.6 Epilogue
2 Sequences and Series
2.1 Discussion: Rearrangements of Infinite Serie
2.2 The Limit of a Sequence
2.3 The Algebraic and Order Limit Theorems
2.4 The Monotone Convergence Theorem and a First Look at Infinite Series
2.5 Subsequences and the Bolzano-Weierstrass Theorem
2.6 The Cauchy Criterion
2.7 Properties of Infinite Series
2.8 Double Summations and Products of Infinite Series
2.9 Epilogue
3 Basic Topology of R
3.1 Discussion: The Cantor Set
3.2 Open and Closed Sets
3.3 Compact Sets
3.4 Perfect Sets and Connected Sets
3.5 Baire's Theorem
3.6 Epilogue
4 Functional Limits and Continuity
4.1 Discussion: Examples of Dirichlet and Thomae
4.2 Functional Limits
4.3 Combinations of Continuous Functions
4.4 Continuous Functions on Compact Sets
4.5 The Intermediate Value Theorem
4.6 Sets of Discontinuity
4.7 Epilogue The Derivative
5 Discussion: Are Derivatives
5.1 Discussion: Are Derivatives Continuous?
5.2 Derivatives and the Intermediate Value Property
5.3 The Mean Value Theorem
5.4 A Continuous Nowhere-Differentiable Function
5.5 Epilogue
6 Sequences and Series of Functions
6.1 Discussion: Branching Processes
6.2 Uniform Convergence of a Sequence of Functions
6.3 Uniform Convergence and Differentiation
6.4 Series of Functions
6.5 Power Series
6.6 Taylor Series
6.7 Epilogue The Riemann Integral
7 Epilogue The Riemann Integral
7.1 Discussion: How Should Integration he Defined?
7.2 The Definition of the Riemann Integral
7.3 Integrating Functions with Discontinuities
7.4 Properties of the Integral
7.5 The Fundamental Theorem of Calculus
7.6 Lebesgue's Criterion for Riemann Integrability
7.7 Epilogue
8 Additional Topics
8.1 The Generalized Riemann Integral
8.2 Metric Spaces and the Baire Category Theorem
8.3 Fourier Series
8.4 A Construction of R From Q
Bibliography
Index
