1 Real Numbers
1 Preliminaries
2 Cuts
3 Euclidean Space
4 Cardinality
5* Comparing Cardinalities
6* The Skeleton of Calculus
Exercises
2 A Taste of Topology
1 Metric Space Concepts
2 Compactness
3 Connectedness
4 Coverings
5 Cantor Sets
6* Cantor Set Lore
7* Completion
Exercises
3 Functions of a Real Variable
1 Differentiation
2 Riemann Integration
3 Series
Exercises
4 Function Spaces
1 Uniform Convergence and C0[a, b]
2 Power Series
3 Compactness and Equicontinuity in CO
4 Uniform Approximation in Co
5 Contractions and ODE's
6* Analytic Functions
7* Nowhere Differentiable Continuous Functions
8* Spaces of Unbounded Functions
Exercises
5 Multivariable Calculus
1 Linear Algebra
2 Derivatives
3 Higher derivatives
4 Smoothness Classes
5 Implicit and Inverse Functions
6* The Rank Theorem
7* Lagrange Multipliers
8 Multiple Integrals
9 Differential Forms
10 The General Stokes' Formula
11* The Brouwer Fixed Point Theorem
Appendix A: Perorations of Dieudonne
Appendix B: The History of Cavalieri's Principle
Appendix C: A Short Excursion into
the Complex Field
Appendix D: Polar Form
Appendix E: Determinants
Exercises
6 Lebesgue Theory
1 Outer measure
2 Measurability
3 Regularity
4 Lebesgue integrals
5 Lebesgue integrals as limits
6 Italian Measure Theory
7 Vitali coverings and density points
8 Lebesgue's Fundamental Theorem of Calculus
9 Lebesgue's Last Theorem
Appendix A: Translations and Nonmeasurable sets
Appendix B: The Banach-Tarski Paradox
Appendix C: Riemann integrals as undergraphs
Appendix D: Littlewood's Three Principles
Appendix E: Roundness
Appendix F: Money
Suggested Reading
Bibliography
Exercises
Index
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