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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