A Primer of Real Analysis

by Dan Sloughter

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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.

Description from LibreTexts.

Publisher
Furman University
Subjects
Mathematics, Analysis
Licence
CC BY-NC-SA, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 1: Fundamentals
  7. 1.1: Sets and Relations
  8. 1.2: Functions
  9. 1.3: Rational Numbers
  10. 1.4: Real Numbers
  11. 2: Sequences and Series
  12. 2.1: Sequences
  13. 2.2: Infinite Series
  14. 3: Cardinality
  15. 3.1: Binary Representations
  16. 3.2: Countable and Uncountable Sets
  17. 3.3: Power Sets
  18. 4: Topology of the Real Line
  19. 4.1: Intervals
  20. 4.2: Open Sets
  21. 4.3: Closed Sets
  22. 4.4: Compact Sets
  23. 5: Limits and Continuity
  24. 5.1: Limits
  25. 5.2: Monotonic Functions
  26. 5.3: Limits to Infinity and Infinite Limits
  27. 5.4: Continuous Functions
  28. 6: Derivatives
  29. 6.1: Best Linear Approximations
  30. 6.2: Derivatives
  31. 6.3: Mean Value Theorem
  32. 6.4: Discontinuities of Derivatives
  33. 6.5: L'Hopital's Rule
  34. 6.6: Taylor's Theorem
  35. 7: Integrals
  36. 7.1: Upper and Lower Integrals
  37. 7.2: Integrals
  38. 7.3: Integrability Conditions
  39. 7.4: Properties of Integrals
  40. 7.5: The Fundamental Theorem of Calculus
  41. 7.6: Taylor's Theorem Revisited
  42. 7.7: An Improper Integral
  43. 8: More Functions
  44. 8.1: The Arctangent Function
  45. 8.2: The Tangent Function
  46. 8.3: The Sine and Cosine Functions
  47. 8.4: The Logarithm Functions
  48. 8.5: The Exponential Function
  49. Back Matter
  50. Index
  51. Glossary
  52. Detailed Licensing