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A Primer of Real Analysis
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.
- Author
- Dan Sloughter
- Publisher
- Furman University
- Source
- LibreTexts
- License
- CC BY-NC-SA
- Language
- EN
- Category
- Basic Sciences
Contents· 43 sections
- 01: Fundamentals
- 1.01: Sets and Relations
- 1.02: Functions
- 1.03: Rational Numbers
- 1.04: Real Numbers
- 02: Sequences and Series
- 2.01: Sequences
- 2.02: Infinite Series
- 03: Cardinality
- 3.01: Binary Representations
- 3.02: Countable and Uncountable Sets
- 3.03: Power Sets
- 04: Topology of the Real Line
- 4.01: Intervals
- 4.02: Open Sets
- 4.03: Closed Sets
- 4.04: Compact Sets
- 05: Limits and Continuity
- 5.01: Limits
- 5.02: Monotonic Functions
- 5.03: Limits to Infinity and Infinite Limits
- 5.04: Continuous Functions
- 06: Derivatives
- 6.01: Best Linear Approximations
- 6.02: Derivatives
- 6.03: Mean Value Theorem
- 6.04: Discontinuities of Derivatives
- 6.05: L'Hopital's Rule
- 6.06: Taylor's Theorem
- 07: Integrals
- 7.01: Upper and Lower Integrals
- 7.02: Integrals
- 7.03: Integrability Conditions
- 7.04: Properties of Integrals
- 7.05: The Fundamental Theorem of Calculus
- 7.06: Taylor's Theorem Revisited
- 7.07: An Improper Integral
- 08: More Functions
- 8.01: The Arctangent Function
- 8.02: The Tangent Function
- 8.03: The Sine and Cosine Functions
- 8.04: The Logarithm Functions
- 8.05: The Exponential Function