To the Instructor
To the Student
Introduction and Outline of the Book
Acknowledgments
Part 1. Rebuilding the Calculus Building
Chapter 1. The Real Number System Revisited
1.1. The Algebraic Axioms
1.2. The Order Axioms
1.3. Absolute Value, Distance, and Neighborhoods
1.4. Natural Numbers and Mathematical Induction
1.5. The Axiom of Completeness and Its Uses
1.6. The Complex Number System
Notes on Essence and Generalizability
Exercises
Chapter 2. Sequences and Series of Real Numbers
2.1. Real Sequences, Their Convergence, and Boundedness
2.2. Subsequences, Limit Superior and Limit Inferior
2.3. Cauchy Sequences
2.4. Sequences in Closed and Bounded Intervals
2.5. Series: Revisiting Some Convergence Tests
2.6. Rearrangements of Series
2.7. Power Series
Notes on Essence and Generalizability
Exercises
Chapter 3. Limit and Continuity of Real Functions
3.1. Limit Points and Some Other Classes of Points in ℝ
3.2. A More General Definition of Limit
3.3. Limit at Infinity
3.4. One-Sided Limits\index{one-sided!limit}
3.5. Continuity and Two Kinds of Discontinuity
3.6. Continuity on [𝑎, 𝑏]: Results and Applications
3.7. Uniform Continuity
Notes on Essence and Generalizability
Exercises
Chapter 4. Derivative and Differentiation
4.1. The Why and What of the Concept of Derivative
4.2. The Basic Properties of Derivative
4.3. Local Extrema and Derivative
4.4. The Mean Value Theorem: More Applications of Derivative
4.5. Taylor Series: A First Glance
4.6. Taylor’s Theorem and the Convergence of Taylor Series
Notes on Essence and Generalizability
Exercises
Chapter 5. The Riemann Integral
5.1. Motivation: The Area Problem
5.2. The Riemann Integral: Definition and Basic Results
5.3. Some Integrability Theorems
5.4. Antiderivatives and the Fundamental Theorem of Calculus
Notes on Essence and Generalizability
Exercises
Part 2. Abstraction and Generalization
Chapter 6. Basic Theory of Metric Spaces
6.1. A First Generalization: The Definition of Metric Space
6.2. Neighborhoods and Some Classes of Points
6.3. Open and Closed Sets
6.4. Metric Subspaces
6.5. Boundedness and Total Boundedness
Notes on Essence and Generalizability
Exercises
Chapter 7. Sequences in General Metric Spaces
7.1. Convergence and Divergence in Metric Spaces
7.2. Cauchy Sequences and Complete Metric Spaces
7.3. Compactness: Definition and Some Basic Results
7.4. Compactness: Some Equivalent Forms
7.5. Perfect Sets and Cantor’s Set
Notes on Essence and Generalizability
Exercises
Chapter 8. Limit and Continuity of Functions in Metric Spaces
8.1. The Definition of Limit in General Metric Spaces
8.2. Continuity and Uniform Continuity
8.3. Continuity and Compactness
8.4. Connectedness and Its Relation to Continuity
8.5. Banach’s Fixed Point Theorem
Notes on Essence and Generalizability
Exercises
Chapter 9. Sequences and Series of Functions
9.1. Sequences of Functions and Their Pointwise Convergence
9.2. Uniform Convergence
9.3. Weierstrass’s Approximation Theorem
9.4. Series of Functions and Their Convergence
Notes on Essence and Generalizability
Exercises
Appendix
Real Sequences and Series
Limit and Continuity of Functions
The Concepts of Derivative and Differentiability
The Riemann Integral
Bibliography
Index