## A Primer of Real FunctionsThis is a revised, updated, and augmented edition of a classic Carus monograph with a new chapter on integration and its applications. Earlier editions covered sets, metric spaces, continuous functions, and differentiable functions. To that, this edition adds sections on measurable sets and functions and the Lebesgue and Stieltjes integrals. The book retains the informal chatty style of the previous editions. It presents a variety of interesting topics, many of which are not commonly encountered in undergraduate textbooks, such as the existence of continuous everywhere-oscillating functions; two functions having equal derivatives, yet not differing by a constant; application of Stieltjes integration to the speed of convergence of infinite series. For readers with a background in calculus, the book is suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Students of mathematics will find here the sense of wonder that was associated with the subject in its early days. |

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### Contents

Sets | 1 |

Sets of real numbers | 5 |

Countable and uncountable sets | 8 |

Metric spaces | 21 |

Open and closed sets | 25 |

Dense and nowhere dense sets | 38 |

Compactness | 45 |

Convergence and completeness | 52 |

Approximations to continuous functions | 126 |

Linear functions | 132 |

Derivatives | 139 |

Monotonic functions | 158 |

Convex functions | 175 |

Infinitely differentiable functions | 186 |

Integration | 195 |

Measurable functions | 201 |

Nested sets and Baires theorem | 61 |

Some applications of Baires Theorem | 66 |

Sets of measure zero | 73 |

Functions | 77 |

Continuous functions | 83 |

Properties of continuous functions | 90 |

Upper and lower limits | 105 |

Sequences of functions | 108 |

Uniform convergence | 112 |

Pointwise limits of continuous functions | 123 |

### Common terms and phrases

American Mathematical Monthly Baire's theorem boundary points Cantor set chord of length closed interval closed sets compact complement consider consists construct contains points continuous function converges uniformly countable set covered decimal defined definition denote dense set digits Dini derivates discontinuous disjoint domain elements empty endpoints equal everywhere dense example Exercise exists finite number func function g graph Hence horizontal chord interior point intermediate value property intersection interval 0,1 interval in Ri inverse image least upper bound Lebesgue integral lim sup limit point linear maximum mean-value theorem measurable functions measure zero metric space monotonic function nondecreasing one-to-one correspondence open intervals open sets partial sums pointwise polynomial positive integers positive number proof prove rational numbers real numbers Riemann integral right-hand set in Ri set of measure set of points Stieltjes integral subinterval subset suppose supremum tion triangle inequality uncountable uniformly convergent union