Read more about A Primer of Real Analysis

A Primer of Real Analysis

(1 review)

Dan Sloughter, Furman University

Copyright Year: 2009

Publisher: Dan Sloughter

Language: English

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Reviewed by Seonguk Kim, Assistant of Professor of Mathematics, DePauw University on 9/20/19

This book consists of all essential sections that students should know in the class, Analysis or Introduction of Real Analysis. First, in chapter 1, it has crucial prerequisite contents. Second, from chapter 2 to 8, the order of sections is... read more

Table of Contents

1 Fundamentals

  • 1.1 Sets and relations
  • 1.2 Functions
  • 1.3 Rational numbers
  • 1.4 Real Numbers

2 Sequences and Series

  • 2.1 Sequences
  • 2.2 Infinite series

3 Cardinality

  • 3.1 Binary representations
  • 3.2 Countable and uncountable sets
  • 3.3 Power sets

4 Topology of the Real Line

  • 4.1 Intervals
  • 4.2 Open sets
  • 4.3 Closed sets
  • 4.4 Compact Sets

5 Limits and Continuity

  • 5.1 Limits
  • 5.2 Monotonic functions
  • 5.3 Limits to infinity and infinite limits
  • 5.4 Continuous Functions

6 Derivatives

  • 6.1 Best linear approximations
  • 6.2 Derivatives
  • 6.3 Mean Value Theorem
  • 6.4 Discontinuities of derivatives
  • 6.5 l'Hˆopital's rule
  • 6.6 Taylor's Theorem

7 Integrals

  • 7.1 Upper and lower integrals
  • 7.2 Integrals
  • 7.3 Integrability conditions
  • 7.4 Properties of integrals
  • 7.5 The Fundamental Theorem of Calculus
  • 7.6 Taylor's theorem revisited
  • 7.7 An improper integral

8 More Functions

  • 8.1 The arctangent function
  • 8.2 The tangent function
  • 8.3 The sine and cosine Functions
  • 8.4 The logarithm function
  • 8.5 The exponential function

Index

About the Book

This is a short introduction to the fundamentals of real analysis. Although the prerequisites are few, I have written the text assuming the reader has the level of mathematical maturity of one who has completed the standard sequence of calculus courses, has had some exposure to the ideas of mathematical proof (including induction), and has an acquaintance with such basic ideas as equivalence relations and the elementary algebraic properties of the integers.

About the Contributors

Author

Dan Sloughter has been teaching Furman students since 1986, and became Professor of Mathematics in 1996. He previously served as an assistant professor at Santa Clara University from 1983-86, and at Boston College from 1981-83. He was also an instructor at Dartmouth College from 1979-81.