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Introduction to Mathematical Analysis I
Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.
- Author
- Lafferriere, Lafferriere, and Nguyen
- Publisher
- LibreTexts
- Source
- LibreTexts
- License
- CC BY-NC
- Language
- EN
- Category
- Basic Sciences
Contents· 35 sections
- 01: Tools for Analysis
- 1.1: Basic Concepts of Set Theory
- 1.2: Functions
- 1.3: The Natural Numbers and Mathematical Induction
- 1.4: Ordered Field Axioms
- 1.5: The Completeness Axiom for the Real Numbers
- 1.6: Applications of the Completeness Axiom
- 02: Sequences
- 2.1: Convergence
- 2.2: Limit Theorems
- 2.3: Monotone Sequences
- 2.4: The Bolazno-Weierstrass Theorem
- 2.5: Limit Superior and Limit Inferior
- 2.6: Open Sets Closed Sets Compact Sets and Limit Points
- 03: Limits and Continuity
- 3.1: Limits of Functions
- 3.2: Limit Theorems
- 3.3: Continuity
- 3.4: Properties of Continuous Functions
- 3.5: Uniform Continuity
- 3.6: Limit Superior and Limit Inferior of Functions
- 3.7: Lower Semicontinuity and Upper Semicontinuity
- 04: Differentiation
- 4.1: Definition and Basic Properties of the Derivative
- 4.2: THE MEAN VALUE THEOREM
- 4.3: SOME APPLICATIONS OF THE MEAN VALUE THEOREM
- 4.4: LHopitals Rule
- 4.5: Taylors Theorem
- 4.6: CONVEX FUNCTIONS AND DERIVATIVES
- 4.7: NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS
- 05: Solutions and Hints for Selected Exercises
- 5.1: CHAPTER 1
- 5.2: CHAPTER 2
- 5.3: CHAPTER 3
- 5.4: CHAPTER 4