## A Guide to Real VariablesThe purpose of A Guide to Real Variables is to provide an aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated. The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too. |

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### Contents

Sequences | 13 |

Series | 23 |

The Topology of the Real Line | 41 |

Limits and the Continuity of Functions | 55 |

The Derivative | 71 |

The Integral | 85 |

Sequences and Series of Functions | 103 |

Advanced Topics | 115 |

Glossary of Terms from Real Variable Theory | 129 |

141 | |

About the Author | 147 |

### Common terms and phrases

absolutely convergent accumulation point boundary points bounded variation calculate called Cantor set cardinality Cauchy criterion Cauchy sequence closed interval closed set compact set conclude continuous function continuously differentiable Corollary countable define Definition denote dense derivative differentiable function discontinuity element equibounded Example exists f and g Figure finite subcovering fj(x function with domain functions fj converge infimum interval a,b inverse image isolated point Lemma Let f lim f(x limit function limit supremum lower bound Mathematical Mean Value Theorem metric space monotonically increasing function natural numbers non-empty number system one-to-one open covering open interval open set partial sums partition polynomials Proposition Ratio Test rational numbers real analysis real numbers result Riemann integrable Riemann sum Riemann-Stieltjes integral Root Test Ross Honsberger sequence of functions sequence of partial sequence xj series converges series of functions sinx subset supremum uncountable uniform convergence uniformly continuous unit interval Weierstrass write x-+P