## Math 447. Real Variables

Instructor Syllabus

**Text: **Elementary Analysis: *The Theory of Calculus by Kenneth Ross, Second Edition *(2013), ISBN: 978-1461462705, Springer

Chapter | Class Hours | |

1. | Introduction (Omit Section 6: A Development of the Reals) (After Section 4: Completeness, the instructor might add enough material on countability to discuss the uncountability of the reals. Countability is not covered later in the text.) |
4 |

2. | Sequences (About 2 hours should be devoted to Section 13: Some Topological Concepts in Metric Spaces; Omit Section 15: Alternating Series and the Integral Test) |
12 |

3. | Continuity (About 3 hours should be devoted to Section 21: More on Metric Spaces: Continuity and Section 22: More on Metric Spaces: Connectedness) |
9 |

4. | Sequences and Series of Functions (Omit Section 27: Weierstrass Approximation) |
7 |

5. | Differentiation (Omit Section 30: L'Hopital's Rule) (Omit Section 31: Taylor's Theorem) |
3 |

6. | Integration (Omit Section 35: Riemann-Stieltjes Integral) (Omit Section 36: Improper Integrals) (Omit Section 37: Exponents and Logarithms) |
5 |

Leeway and Exams | 3 | |

Total | 43 |

Math 444 and 447 are the first theoretical courses in real analysis offered here. Ideally, a student should leave such a course with not only a basic understanding of the major concepts of the one-dimensional calculus, but an increased facility at reading and writing mathematical sentences, the ability to produce and analyze examples, and some skills at constructing and writing elementary proofs. Math 447 covers enough metric topology to provide background for other courses such as 448. Students in Math 444 will have more time to investigate examples and master the fundamental concepts.

Revised Jan 27, 2015