Real Analysis

IGCSE

IB

USA

Singapore

Primary Year

Secondary Year

Junior College

Poly Technique

College

Course Content

1: PRELIMINARIES

1.1 Sets
1.2 Algebra of Sets
1.3 Indexed sets
1.4 Functions
1.5 Mathematical Induction
1.6 Finite and Infinite Sets

2: THE REAL NUMBERS

3: SEQUENCES AND SERIES

4: LIMITS

5: CONTINUOUS FUNCTIONS

6: DIFFERENTIATION

7: THE RIEMANN INTEGRAL

8: SEQUENCES OF FUNCTIONS

9: INFINITE SERIES

10: THE GENERALIZED RIEMANN INTEGRAL

11: A GLIMPSE INTO TOPOLOGY

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Price : Free
Enrolled :0 Students
Duration :
Lessons :53 Lessons
Quiz :0 Quizzes
Skill Level :Intermediate
Category :College

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IGCSE

Primary Year

Secondary Year

Junior College

Poly Technique

Course Content

1: PRELIMINARIES

1.1 Sets

1.2 Algebra of Sets

1.3 Indexed sets

1.4 Functions

1.5 Mathematical Induction

1.6 Finite and Infinite Sets

2: THE REAL NUMBERS

2.1 The Algebraic and Order Properties of R

2.2 Absolute Value and the Real Line

2.3 The Completeness Property of R

2.4 Applications of the Supremum Property

2.5 Intervals

3: SEQUENCES AND SERIES

3.1 Sequences and Their Limits

3.2 Limit Theorems

3.3 Monotone Sequences

3.4 Subsequences and the Bolzano-Weierstrass Theorem

3.5 The Cauchy Criterion

3.6 Lower and Upper Limits of a sequence

3.7 Properly Divergent Sequences

3.8 Introduction to Infinite Series

4: LIMITS

4.1 Limits of Functions

4.2 Limit Theorems

4.3 Some Extensions of the Limit Concept

5: CONTINUOUS FUNCTIONS

5.1 Continuous Functions

5.2 Combinations of Continuous Functions

5.3 Continuous Functions on Intervals

5.4 Uniform Continuity

5.5 Continuity and Gauges

5.6 Monotone and Inverse Functions

6: DIFFERENTIATION

6.1 The Derivative

6.2 The Mean Value Theorem

6.3 L’Hôpital’s Rules

6.4 Taylor’s Theorem

7: THE RIEMANN INTEGRAL

7.1 Riemann Integral

7.2 Riemann Integrable Functions

7.3 Fundamental Theorem

7.4 The Darboux Integral

7.5 Approximate Integration

8: SEQUENCES OF FUNCTIONS

8.1 Pointwise and Uniform Convergence

8.2 Interchange of Limits

8.3 The Exponential and Logarithmic Functions

8.4 The Trigonometric Functions

9: INFINITE SERIES

9.1 Absolute Convergence

9.2 Tests for Absolute Convergence

9.3 Tests for Non-absolute Convergence

9.4 Series of Functions

10: THE GENERALIZED RIEMANN INTEGRAL

10.1 Definition and Main Properties

10.2 Improper and Lebesgue Integrals

10.3 Infinite Intervals

10.4 Convergence Theorems

11: A GLIMPSE INTO TOPOLOGY

11.1 Open and Closed Sets in R

11.2 Compact Sets

11.3 Continuous Functions

11.4 Metric Spaces

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