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### Introduction to Real Analysis (FOR DEGREE HONOURS COURSE) by Sadhan Kumar Mapa

MathSchoolinternational.com contain houndreds of Free Math e-Books. Which cover almost all topics of mathematics. To see an extisive list of Real Analysis eBooks . We hope mathematician or person who’s interested in mathematics like these books. Introduction to Real Analysis (FOR DEGREE HONOURS COURSE) written by Sadhan Kumar MapaReader in Mathematics (retired), Presidency College, Calcutta . This is an other book of mathematics cover the following topics.

• l. Set Theory
1.1. Introduction
1 1.2. Subsets
1 1.3. Algebraic operations on sets
2 1.4. Family of sets
1.5. Cartesian product of sets
4 5 1.6. Relation on a set . ~
5 1.7. Order relation on a set
6 1.8. Function
1.9. Equipotent sets, enumerable set
10 Exercises 1

• 2. Real numbers
2.l. Natural numbers
2.2. Integers
2.3. Rational numbers
2.4. Real numbers
Exercises 2

• 3. Sets in R
3.1. Intervals
3.2. Neighbourhood
3.3. Interior point
3.4. Open set
3.5. Limit point
3.6. Isolated point
3.7. Derived set
3.8. Closed set
3.10. Dense set, Perfect set
Exercises 3
3.11. Nested intervals
3.12. Decimal representation
3.13. Enumerable set
Exercises 4
3.14. Point of condensation
3,15. Borel set 85
3.16. Cover,. open cover
Exercises 5

• 4. Real functions
4.1. Real function
4.2. Injective function, Surjective function
4.3. Equal functions
4.4. Restriction function
4.5. Composite function
4.6. Inverse function
4.7. Algebraic operations on functions
4.8. Monotone functions
4.9. Even function; odd function
4.10. Power function
4.11. Exponential fUnction
4.12. Logarithmic function
4.13. Hyperbolic functions
4.14. Bounded function
Exercises 6

• 5. Sequence
5.1. Real sequence
5.2. Bounded sequence
5.3. Limit of a sequence
5.4. Convergent sequence
5.5. Limit theorems
5.6. Null sequence
5.7. Divergent sequence
5.8. Some important limits
5.9. Monotone sequence
5.10. Some important sequences
Exercises 7
5.11. Subsequence
5.12. Subsequential limit
5.13. Characterisation of a compact set
5.14. Upper limit and lower limit
5.15. Cauchy criterion
5.16. Cauchy's theorems on limits
Exercises 8

• 6. Series
6.1. Infinite series
6.2. Series of positive terms
6.3. Tests for convergence
Exercises 9
6.4. Series of arbitrary terms
6.5. Conditionally convergent series
6.6. Multiplication of series
Exercises 10

• 7. Limits
7.1. Limit of a function
7.2. One-sided limits
7.3. Infinite limits
7.4. Limits at infinity
7.5. Infinite limits at infinity
7.6. Limits of monotone functions
7.7. Some important'limits
Exercises 11

• 8. Continuity
8.1. Continuity
8.2. Continuity of some important functions
8.3. Limit of composite functions
8.4. Discontinuity
Exercises 12
8.5. Properties of continuous functions
8.6. Monotone functions and continuity
8.7. Uniform continuity
8.8. Continuity on a compact set
Exercises 13

• 9. Differentiation
9.1. Differentiability. Derivative
9.2. Higher order derivatives
Exercises 14
9.3. Sign of the derivative
9.4. Properties of the derivative
9.5. Rolle's theorem and Mean value theorems
Exercises 15
9.6. The nth order derivatives
Exercises 16
9.7. Taylor's theorem and expansion of functions
Exercises 17
9.8. Maxima and minima
Exercises 18
9.9. Indeterminate forms
Exercises 19

• 10. Functions of bounded variation
10.1. Introduction
10.2. Variation function
10.3. Positive variation, negative variation
Exercises 20

• 11. Riemann integral
11.1. Partition
11.2. Riemann integrability
11.3. Refinement of a partition
11.4. Norm of a partition
11.5. Some Riemann integrable functions
11.6. Properties of Riemann integrable functions
11.7. Inequalities
11.8. Fundamental theorem
11:9. Another definition of integrability
11.10. Integration by substitution
11.11. Integration by parts
11.12. Mean value theorems
11.13. Logarithmic function
11.14. Exponential function
Exercises 21

• 12. Improper integrals
12.1. Introduction
12.2. Definitions
12.3. Tests for convergence (positive integrand)
12.4. Tests for convergence
12.5. Definitions
12.6. Tests for convergence (positive integrand)
12.7. Tests for convergence
12.8. Tests for convergence of the integral of a product
12.9. Some theorems
12.10. Evaluation of some improper integrals
Exercises 22
12.11. Beta function and Gamma function ~
Exercises 23

• 13. Sequence of functions
13.1. Sequence of functions
13.2. Pointwise convergence
13.3. Uniform convergence
13.4. Consequences of uniform convergence

• 14 Series of functions
14.1. Uniform convergence
14.2. Consequences of uniform convergence
14.3. Abel's and Dirichlet's tests
Exercises 25

• 15. Power series J .
15.1. Introduction
15.2. Determination of radius of convergence
15.3. Properties of a power series
Exercises 26

• APPENDIX 1- Sets in IR?

• APPENDIX 2- Sequence in IR2

• BIBLIOGRAPHY

• INDEX

• ##### Math Books of Real Analysis

Elements of Real Analysis by Shanti Narayan
• Free
• English
• Page 2121

• Introduction to Real Analysis by Sadhan Kumar Mapa
• Free
• English
• Page 328

• Basic Real Analysis by Anthony W. Knapp
• Free
• English
• PDF 83
• Page 840

• Guide to Analysis by F. Mary Hart
• Free
• English
• Page 213

• The Continuum (1E) by Rudolf Taschner
• Free
• English
• PDF 34
• Page 142
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##### Worksheets (Solved)

###### Integration     