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Basic and Real Analysis
Real Analysis is a rigorous and foundational course in advanced mathematics that delves into the study of real numbers, functions, and their properties. Real Analysis provides the tools and techniques necessary for a deeper understanding of calculus, as well as for the exploration of more advanced mathematical subjects. This course aims to equip students with a solid theoretical foundation and problemsolving skills essential for further studies in mathematics, physics, engineering, and related disciplines.
Course Objectives:
 Develop a rigorous understanding of the real number system, including properties of real numbers, order relations, and completeness.
 Study the concepts of limits, continuity, and differentiability of functions, and explore their applications in calculus.
 Investigate the convergence and divergence of sequences and series, and analyze their properties.
 Introduce the theory of integration, including Riemann integration and its fundamental properties.
 Explore the fundamental concepts of metric spaces and topological spaces, and their relevance in analysis.
 Examine the properties of continuous functions and their implications on connectedness and compactness.
 Introduce the concept of differentiation in higher dimensions, including partial derivatives and the chain rule.
 Study the theory of infinite series and power series expansions, and analyze their convergence and differentiability.
 Investigate the fundamental theorems of calculus, such as the mean value theorem and the fundamental theorem of calculus.
 Develop proficiency in constructing rigorous mathematical proofs and logical reasoning.
Sequences

1Introduction

2Sequences and limits

3Bounded sequences and unique limits

4Theorem on limits

5Sandwich theorem

6Supremum and Infimum

7Cauchy sequences and Completeness

8Example Calculation

9Subsequences and accumulation values

10BolzanoWeierstrass theorem

11Limit superior and limit inferior

12Examples for Limit superior and limit inferior

13Open, Closed and Compact Sets

14HeineBorel theorem
Series
Functions

23Sequence of Functions

24Pointwise Convergence

25Uniform Convergence

26Limits for Functions

27Continuity and Examples

28EpsilonDelta Definition

29Combination of Continuous Functions

30Continuous Images of Compact Sets are Compact

31Uniform Limits of Continuous Functions are Continuous

32Intermediate Value Theorem

33Some Continuous Functions
Differentiation

34Differentiability

35Properties for Derivatives

36Chain Rule

37Uniform Convergence for Differentiable Functions

38Examples of Derivatives and Power Series

39Derivatives of Inverse Functions

40Local Extrema and Rolle's Theorem

41Mean Value Theorem

42L'Hôpital's Rule

43Other L'Hôpital's Rules

44Higher Derivatives

45Taylor's Theorem

46Application for Taylor's Theorem

47Proof of Taylor's Theorem
Integration

48Riemann Integral  Partitions

49Riemann Integral for Step Functions

50Properties of the Riemann Integral for Step Functions

51Riemann Integral  Definition

52Riemann Integral  Examples

53Riemann Integral  Properties

54First Fundamental Theorem of Calculus

55Second Fundamental Theorem of Calculus

56Proof of the Fundamental Theorem of Calculus

57Integration by Substitution

58Integration by Parts

59Integration by Partial Fraction Decomposition

60Integrals on Unbounded Domains

61Comparison Test for Integrals

62Integral Test for Series

63Improper RiemannIntegrals for Unbounded Functions

64Cauchy Principal Value
Examination Questions
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