## Real Mathematical AnalysisWas plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and I hope it appeals to you, the budding pure mathematician. Berkeley, California, USA CHARLES CHAPMAN PUGH Contents 1 Real Numbers 1 1 Preliminaries 1 2 Cuts . . . . . 10 3 Euclidean Space . 21 4 Cardinality . . . 28 5* Comparing Cardinalities 34 6* The Skeleton of Calculus 36 Exercises . . . . . . . . 40 2 A Taste of Topology 51 1 Metric Space Concepts 51 2 Compactness 76 3 Connectedness 82 4 Coverings . . . 88 5 Cantor Sets . . 95 6* Cantor Set Lore 99 7* Completion 108 Exercises . . . 115 x Contents 3 Functions of a Real Variable 139 1 Differentiation. . . . 139 2 Riemann Integration 154 Series . . 179 3 Exercises 186 4 Function Spaces 201 1 Uniform Convergence and CO[a, b] 201 2 Power Series . . . . . . . . . . . . 211 3 Compactness and Equicontinuity in CO . 213 4 Uniform Approximation in CO 217 Contractions and ODE's . . . . . . . . 228 5 6* Analytic Functions . . . . . . . . . . . 235 7* Nowhere Differentiable Continuous Functions . 240 8* Spaces of Unbounded Functions 248 Exercises . . . . . 251 267 5 Multivariable Calculus 1 Linear Algebra . . 267 2 Derivatives. . . . 271 3 Higher derivatives . 279 4 Smoothness Classes . 284 5 Implicit and Inverse Functions 286 290 6* The Rank Theorem 296 7* Lagrange Multipliers 8 Multiple Integrals . . |

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Great introductory book, especially for people who wish to self-study Real Analysis. Note that for any high school students who wish to self-study from this book should have a strong background in Mathematics. General understanding of the Propositional Calculus (a.k.a. Mathematical Logic) and Set Theory will help in the beginning, but once you get through the first chapter, the rest of the book slowly becomes accessible.

### Contents

I | 1 |

II | 10 |

III | 21 |

IV | 28 |

V | 34 |

VI | 36 |

VII | 40 |

VIII | 51 |

XXXII | 290 |

XXXIII | 296 |

XXXIV | 300 |

XXXV | 313 |

XXXVI | 325 |

XXXVII | 334 |

XXXVIII | 337 |

XXXIX | 338 |

IX | 76 |

X | 82 |

XI | 88 |

XII | 95 |

XIII | 99 |

XIV | 108 |

XV | 139 |

XVI | 154 |

XVII | 179 |

XVIII | 186 |

XIX | 201 |

XX | 211 |

XXI | 213 |

XXII | 217 |

XXIII | 228 |

XXIV | 235 |

XXV | 240 |

XXVI | 248 |

XXVII | 267 |

XXVIII | 271 |

XXIX | 279 |

XXX | 284 |

XXXI | 286 |

### Common terms and phrases

algebra analytic Assume that f bijection Calculus Cantor set Cantor space Cauchy sequence clopen closed set closure complete metric space constant contains continuous function converges uniformly convex Corollary definition dense denumerable derivative Df)p diffeomorphism differentiable discontinuity disjoint dyadic equal equicontinuous equivalent everywhere example Exercise exists f is continuous Figure finite fixed point formula function defined function f given gives graph Hint homeomorphism implies infinite intersection inverse least upper bound Lebesgue integral Lemma Let f limit linear transformation Mathematics matrix Mean Value Theorem measurable set monotone neighborhood non-empty norm open intervals open set outer measure partition plane pointwise polynomial power series pre-image Proof Let Prove that f rational numbers real numbers rectangles Riemann integrable Suppose topology uncountable undergraph uniformly continuous variable vector space Vitali Covering zero set