This volume studies functions of real variables with an emphasis on limits, continuity, differentiation, integration, and fine properties of...
This volume studies functions of real variables with an emphasis on limits, continuity, differentiation, integration, and fine properties of functions. It builds the rigorous core of real analysis.
Part I. Real Numbers and Limits
Chapter 1. The Real Number System
1.1 Ordered fields 1.2 Completeness axiom 1.3 Suprema and infima 1.4 Sequences in ℝ 1.5 Convergence
Chapter 2. Limits of Sequences
2.1 Definition of limit 2.2 Limit laws 2.3 Monotone convergence 2.4 Cauchy sequences 2.5 Subsequences
Chapter 3. Limits of Functions
3.1 ε–δ definition 3.2 One-sided limits 3.3 Infinite limits 3.4 Limit laws 3.5 Examples
Part II. Continuity
Chapter 4. Continuous Functions
4.1 Definition 4.2 Properties 4.3 Composition 4.4 Examples 4.5 Discontinuities
Chapter 5. Properties of Continuous Functions
5.1 Intermediate value theorem 5.2 Extreme value theorem 5.3 Uniform continuity 5.4 Compactness in ℝ 5.5 Applications
Chapter 6. Sequences and Series of Functions
6.1 Pointwise convergence 6.2 Uniform convergence 6.3 Interchange of limits 6.4 Examples 6.5 Applications
Part III. Differentiation
Chapter 7. Derivatives
7.1 Definition 7.2 Rules of differentiation 7.3 Higher derivatives 7.4 Examples 7.5 Applications
Chapter 8. Mean Value Theorems
8.1 Rolle’s theorem 8.2 Mean value theorem 8.3 Consequences 8.4 Applications 8.5 Inequalities
Chapter 9. Taylor Theory
9.1 Taylor polynomials 9.2 Remainder terms 9.3 Approximation 9.4 Analytic functions (overview) 9.5 Applications
Part IV. Integration
Chapter 10. Riemann Integral
10.1 Definitions 10.2 Integrability conditions 10.3 Properties 10.4 Examples 10.5 Applications
Chapter 11. Fundamental Theorem of Calculus
11.1 Statement and proof 11.2 Differentiation of integrals 11.3 Applications 11.4 Examples 11.5 Extensions
Chapter 12. Improper Integrals
12.1 Infinite intervals 12.2 Singular integrals 12.3 Convergence tests 12.4 Applications 12.5 Examples
Part V. Series
Chapter 13. Numerical Series
13.1 Convergence tests 13.2 Absolute and conditional convergence 13.3 Comparison tests 13.4 Power series 13.5 Examples
Chapter 14. Power Series
14.1 Radius of convergence 14.2 Operations 14.3 Differentiation and integration 14.4 Examples 14.5 Applications
Chapter 15. Fourier Series (Overview)
15.1 Periodic functions 15.2 Fourier coefficients 15.3 Convergence issues 15.4 Applications 15.5 Examples
Part VI. Advanced Topics
Chapter 16. Functions of Bounded Variation
16.1 Definitions 16.2 Jordan decomposition 16.3 Properties 16.4 Applications 16.5 Examples
Chapter 17. Absolute Continuity
17.1 Definitions 17.2 Relationship to differentiation 17.3 Fundamental properties 17.4 Applications 17.5 Examples
Chapter 18. Differentiability Properties
18.1 Nowhere differentiable functions 18.2 Lipschitz conditions 18.3 Hölder continuity 18.4 Fine structure 18.5 Examples
Part VII. Functional Perspective
Chapter 19. Function Spaces
19.1 Metric spaces of functions 19.2 Norms and convergence 19.3 Completeness 19.4 Examples 19.5 Applications
Chapter 20. Operators on Functions
20.1 Linear operators 20.2 Integral operators 20.3 Differentiation operators 20.4 Applications 20.5 Examples
Chapter 21. Approximation Theory
21.1 Polynomial approximation 21.2 Weierstrass theorem 21.3 Approximation methods 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Real analysis and measure theory links 22.2 Harmonic analysis connections 22.3 Fractal functions 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Convergence questions 23.2 Differentiability issues 23.3 Approximation challenges 23.4 Functional analysis links 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of real analysis 24.2 Key contributors 24.3 Evolution of rigor 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common limits and identities B. Convergence test summary C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume builds real analysis as the rigorous study of functions on ℝ. It emphasizes limits, continuity, differentiation, and integration as core analytical tools.