Real mathematical analysis / Charles C. Pugh.

Includes bibliographical references (pages 469-470) and index.

1. REAL NUMBERS -- 1. Preliminaries, pages1 -- 2. Cuts, pages11 -- 3. Euclidean Space, pages22 -- 4. Cardinality, pages29 -- 5. Comparing Cardinalities, pages36 -- 6. The Skeleton of Calculus, pages41 -- 7. Visualizing the Fourth Dimension, pages41 -- Exercises, pages44 -- 2. A TASTE OF TOPOLOGY -- 1. Metric Spaces, pages57 -- 2. Continuity, pages61 -- 3. The Topology of a Metric Space, pages65 -- 4. Compactness, pages79 -- 5. Connectedness, pages86 -- 6. Other Metric Space Concepts, pages92 -- 7. Coverings, pages98 -- 8. Cantor Sets, pages105 -- 9. Cantor Set Lore, pages108 -- 10. Completion, pages119 -- Exercises , pages125 -- 3. FUNETIONS OF A REAL VARIABLE -- 1. Differentiation, pages149 -- 2. Riemann Integration, pages164 -- 3. Series, pages191 -- Exercises, pages198 -- 4. FUNCTION SPAEES -- 1. Uniform Convergence and CO[a, bL, pages211 -- 2. Power Series, pages220 -- 3. Compactness and Equicontinuity in CO, pages223 -- 4. Uniform Approximation in CO, pages228 -- 5. Contractions and ODEs, pages240 -- 6. Analytic Functions, pages248 -- 7. Nowhere Differentiable Continuous Functions, pages253 -- 8. Spaces of Unbounded Functions, pages260 -- Exereises, pages263

5. MULTIVARIABLE CALEULUS -- 1. Linear Algebra, pages277 -- 2. Derivatives, pages282 -- 3. Higher Derivatives, pages291 -- 4. Implicit and Inverse Functions, pages297 -- 5. The Rank Theorem, pages301 -- 6. Lagrange Multipliers, pages310 -- 7. Multiple Integrals, pages313 -- 8. Differential Forms, pages326 -- 9. The General Stokes Formula -- 10. The Brouwer Fixed-Point Theorem, P.353 -- Appendix A. Perorations of Dieudonne, pages357 -- Appendix B. The HistoryofCavalieri's Principle, pages358 -- Appendix C. A Short Excursion into the Complex Field, pages359 -- Appendix D. Polar Form, pages360 -- Appendix E. Determinants, pages363 -- Exereises, pages366 -- 6. LEBESGUE THEORY -- 1. Outer Measure, pages383 -- 2. Measurability, pages388 -- 3. Meseomorphism, pages393 -- 4. Regularity, pages397 -- 5. Products and Slices, pages401 -- 6. Lebesgue Integrals, pages406 -- 7. ltalian Measure Theory, pages414 -- 8. Vitali Coverings and Density Points, pages418 -- 9. Calculus a la Lebesgue, pages426 -- 10. Lebesgue's Last Theorem, pages433 -- Appendix A. Lebesgue integrals as limits, pages440 -- Appendix B. Nonmeasurable sets, pages440 -- Appendix C. Borel versus Lebesgue, pages443 -- Appendix D. The Banach- Tarski Paradox, pages444 -- Appendix E. Riemann integrals as undergraphs, pages445 -- Appendix F. Littlewood's Three Principles, pages447 -- Appendix G. Roundness, pages449 -- Appendix H. Money, pages449 -- Exercises, pages450 -- Suggested Reading, pages467 -- Bibliography, pages469 -- Index, pages471