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Preface

Part I. Fundamental Concepts

1. Sets and Functions

1. Sets

2. The Algebra of Sets

3. Functions

4. Countability

2. The Rational Numbers

5. Alebraic Properties

6. Decimal Expansions

Part II. The Real Line

3. The Real Number System

7. Cauchy Sequences and Their Equivalence Classes

8. The Real Number System

9. Completeness Properties of R

10. The Extended Real Line

4. Sequences and Series of Number

11. Sequences: Basic Limit Theorems

12. Upper and Lower Limits

13. Basic Properties of Series

14. Series with Nonnegative Terms

15. Alternating Series

16. Absolute Convergence

5. The Structure of Point Sets

17. Basic Notions

18. Closed Sets

19. Open Sets

20. Perfect Sets

21. Distance between Point Sets

22. Connected Sets

3. Functions of a Real Variable

23. Continuity Limits of Functions

24. Continuous Functions

25. The Nature of Discontinuities

26. Monotonic Functions

27. Uniform Continuity

7. Differentiability

28 The Derivative at a Point

29. A Continuous Nowhere Differentiable Function

30. Properties of the Derivative

31. Taylor's Theorem

8. Spaces of Continuous Functions

32. The Problems of Separability and Convergence

33. Uniform Convergence

34. Power Series

35. The Approximation of Functions

36. Equicontinuity

37. Summary

9. Measure and Integration

38. Measurable Sets

39. Properties of Measurable Sets

40. Measurable Functions

41. The Lebesgue Integral of Simple Functions

42. The Lebesgue Integral

43. Theorems on Limits under the Integral Sign

44. The Riemann Integral

10. Fourier Series

45. Basic Facts

46. The Space £ superscript 2

47. The Question of Convergence

Bibliography Glossary of Symbols Index

Part I. Fundamental Concepts

1. Sets and Functions

1. Sets

2. The Algebra of Sets

3. Functions

4. Countability

2. The Rational Numbers

5. Alebraic Properties

6. Decimal Expansions

Part II. The Real Line

3. The Real Number System

7. Cauchy Sequences and Their Equivalence Classes

8. The Real Number System

9. Completeness Properties of R

10. The Extended Real Line

4. Sequences and Series of Number

11. Sequences: Basic Limit Theorems

12. Upper and Lower Limits

13. Basic Properties of Series

14. Series with Nonnegative Terms

15. Alternating Series

16. Absolute Convergence

5. The Structure of Point Sets

17. Basic Notions

18. Closed Sets

19. Open Sets

20. Perfect Sets

21. Distance between Point Sets

22. Connected Sets

3. Functions of a Real Variable

23. Continuity Limits of Functions

24. Continuous Functions

25. The Nature of Discontinuities

26. Monotonic Functions

27. Uniform Continuity

7. Differentiability

28 The Derivative at a Point

29. A Continuous Nowhere Differentiable Function

30. Properties of the Derivative

31. Taylor's Theorem

8. Spaces of Continuous Functions

32. The Problems of Separability and Convergence

33. Uniform Convergence

34. Power Series

35. The Approximation of Functions

36. Equicontinuity

37. Summary

9. Measure and Integration

38. Measurable Sets

39. Properties of Measurable Sets

40. Measurable Functions

41. The Lebesgue Integral of Simple Functions

42. The Lebesgue Integral

43. Theorems on Limits under the Integral Sign

44. The Riemann Integral

10. Fourier Series

45. Basic Facts

46. The Space £ superscript 2

47. The Question of Convergence

Bibliography Glossary of Symbols Index

Library of Congress subject headings for this publication: Functions of real variables, Mathematical analysis