ELEMENTS OF
REAL ANALYSIS
[For B.A., B.Sc. and Honours (Mathematics and Physics),
M.A. and M.Sc. (Mathematics) students of various Universities/Institutions
as per UGC Model Carriculum. Also useful for GATE
and various other competitive examinations]
SHANTI NARAYAN
Formerly, Dean of Colleges,
University of Delhi, Delhi.
(Formerly, Principal, Hans Raj College, Delhi)
Revised by
Dr. M.D. Raisinghania
M.Sc., Ph.D.
Formerly, Head of Mathematics Department,
S.D. (Postgraduate) College,
Muzaffarnagar (U.P.)
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© 1965, Shanti Narayan and M.D. Raisinghania
All rights reserved. No part of this publication may be reproduced or copied in any material form (including
photo copying or storing it in any medium in form of graphics, electronic or mechanical means and whether
or not transient or incidental to some other use of this publication) without written permission of the copyright
owner. Any breach of this will entail legal action and prosecution without further notice.
Jurisdiction : All disputes with respect to this publication shall be subject to the jurisdiction of the Courts,
tribunals and forums of New Delhi, India only.
First Edition 1965
Subsequent Editions and Reprints 1966, 69, 74, 76, 79, 80, 83, 85, 87, 89, 92, 95, 98, 2001, 2003, 2007
(Twice), 2009, 2010, 2011, 2012
Fourteenth Revised Edition 2013
ISBN : 81-219-0306-8
Code : 14D 052
PRINTED IN INDIA
By Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi-110 055
and published by S. Chand & Company Ltd., 7361, Ram Nagar, New Delhi -110 055.
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PREFACE TO THE FOURTEENTH EDITION
References to the latest papers of various universities and GATE have been inserted at proper
places. More additional problems have been inserted in almost each chapter of this book. New
topics have been inserted in some chapters. I hope that these changes will make the material of
this book more useful to the reader.
All valuable suggestions for further improvement of the book will be highly appreciated.
M.D. Raisinghania
PREFACE TO THE EIGHTH EDITION
The book originally written, about 40 years ago, has during the intevening period, been revised and
reprinted several times. Due to the new U.G.C. Model syllabus and the demand for more matter from the
students and teachers, a thorough revision of the book was overdue. I very humbly took the challenge of
revising this perfect well-written book of late Shri Shanti Narayan with whom I had personal contact from
1962 onwards. He was my teacher and guiding star in art of writting a book.
I have tried to meet the rapidly changing demands of students interested in self-study and appearing
in various examinations. Accordingly, a large variety of illustrative solved examples have been included
in every chapter. References to the latest papers of various universities and I.A.S. examination have been
inserted at proper places.
The following new chapters have been added in this present edition
• Countability of sets • The Riemann-Stieltjes Integrals • Uniform convergence of sequences and
series of functions • Improper Integrals • Metric spaces
The book, in the present form, is a humble effort to make it more useful to the students and teachers.
I am extremely thankfull to the Managing Director, Shri Ravindra Kumar Gupta,
Shri Navin Joshi, Vice President (Publishing) and Advisor, Shri R.S. Saxena for personal intersest
throughout the preparation of the book. My sincere thanks are also due to Mr. Shishir Bhatnagar of S.
Chand & Company for bringing this book in an excellent form.
All valuable suggestions for further improvement of the book will be highly appreciated
M.D. Raisinghania
Preface to the First Edition
This book is an attempt to present Elements of Real Analysis to under-graduate students,* on the basis of
the University Grants Commission Review Committee report recommendations and several Universities having
provided a course along the lines of these recomendations. This book must not, however, be thought of an
abridged edition of the Author’s “A Course of Mathematical Analysis” for M.A. students.
Chapter I provides description of Set of Real Number as a complete ordered field and no attempt has been
made to construct the Set starting from some Axioms. Chapter II deals with bounds and limit points of sets of
real numbers. Chapter III concerns itself with Real sequences defined as functions on the set of Natural numbers
into the set of Real numbers. This is followed by Chapter IV of Infinite Series dealing with mostly convergence
tests for positive term series as also a test on alternating series. Chapter V deals with the nature of the range of
a real valued continuous function with a closed finite interval as its domain. Chapter VI on Derivability deals
with the rigorous proof of Role’s theorem as also of Lagrange’s theorem. Chapter VIII deals with Riemann
Integrability.
The book contains some examples and exercises meant only to help a proper undertaking of the text. The
book also seeks to adopt comparatively more modern notation for the subject.
It is hoped that this “Elements of Real Analysis” will provide a stimulus for the commencement of the
study of Analysis at undergraduate stage which has already been so much delayed.
March, 1965
SHANTI NARAYAN
(iii)
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CONTENTS
1.
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
2.
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.
2.14.
3.
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
3.10.
3.11.
3.12.
3.13.
3.14.
3.15.
SETS AND FUNCTIONS
...
Introduction
...
Statements
...
Connectives
...
Sets
...
Functions (or mappings)
...
Composite of functions (or product of functions)
...
Inverse function
...
Binary operation
...
Objective questions
...
THE REAL NUMBERS
...
Introduction
...
The set N of natural numbers
...
The set I or Z of integers
...
The set Q of rational numbers
...
The set of real numbers R as a complete ordered field
...
Closed, open, semi-closed and semi-open intervals
...
Set bounded above, set bounded below, l.u.b. (supremum) and g.l.b.
(infimum) of a set. The greatest and smallest members of a set. Bounded
and unbounded sets
...
Order-completeness of the set of real numbers
...
Equivalent descriptions of the order-completeness. Property of the
set of real numbers
...
Explicit statement of the properties of the set of real numbers as a
complete ordered field
...
Some important properties of the system of real numbers
...
The denseness property of the set of real numbers R
...
The modulus (or absolute value) of a real number
...
Arithmetic and geometric continua
...
Objective questions
...
NEIGHBOURHOODS AND LIMIT POINTS OF A SET.
OPEN AND CLOSED SETS
...
Introduction
...
Neighbourhood of a point
...
Properties of neighbourhoods
...
Limit (or accumulation or condensation) point of a set
...
Existence of limit points
...
Open and closed sets
...
Basic theorems concerning families of open and closed sets
...
Illustrations of open sets
...
Illustrations of closed sets
...
Interior point and interior of a set
...
Exterior point and exterior of a set
...
Boundary (or frontier) point and boundary (or frontier) of a set
...
Theorems on interior of a set
...
Adherent point (or a contact point) and closure of a set
...
Theorems on closure of a set
...
1.1–1.9
1.1
1.1
1.2
1.4
1.6
1.7
1.8
1.8
1.9
2.1–2.32
2.1
2.2
2.3
2.4
2.6
2.7
2.7
2.16
2.18
2.20
2.20
2.21
2.26
2.29
2.31
3.1–3.36
3.1
3.1
3.3
3.4
3.8
3.13
3.15
3.18
3.19
3.22
3.22
3.22
3.23
3.25
3.26
(iv)
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3.16.
3.17.
6.
6.1.
6.2.
6.3.
6.4.
Isolated points of a set and discrete set
Dense (or everywhere dense), dense in itself, nowhere dense
(or non-dense) and perfect sets.
Cantor nested interval theorem
Cover (or covering) of a set
Compact set
Properties of a compact set
Objective questions
COUNTABILITY OF SETS
Equivalent sets
Finite and infinite sets
Denumerable (or enumerable or countably infinite), countable and
uncountable sets
Decimal, ternary and binary representation
Cantor set or cantor ternary set
Objective questions
SEQUENCES
Sequence
Bounded and unbounded sequences
Limit point (or cluster point or point of condensation) of a sequence
Convergent sequences. The limit of a sequence
Algebra of convergent sequences
Bounded non-convergent sequences
Some theorems on divergent sequences
Some important theorems on limits
Cauchy (or fundamental) sequences
Convergence of a sequence
Monotonic sequences and their convergence
Subsequences
Limit superior and limit inferior of a sequence
Some theorems on limit superior and limit inferior of a sequence
Objective questions
INFINITE SERIES WITH POSITIVE TERMS
Infinite series, its convergence and sum
A necessary condition for the convergence of an infinite series
Cauchy’s general principle of convergence for series
General test for the convergence of positive term series
6.5.
Two important standard series r n and  (1/ n )
...
6.6
6.6.
6.7.
6.8.
6.9.
6.10.
Comparison tests for the convergence of positive term series
Comparison tests of the first type
Practical comparison tests of the first type
Comparison tests of the second type
Practical comparison tests of the second type
D’Alembert’s ratio test
Cauchy’s nth root test
Cauchy’s nth root test is superior than D’Alembert’s ratio test
Raabe’s test
Logarithmic test
De Morgan’s and Bertrand’s test
...
...
...
...
...
...
...
...
...
...
6.8
6.9
6.9
6.13
6.14
6.14
6.19
6.22
6.23
6.25
6.25
3.18.
3.19.
3.20.
3.21.
4.
4.1.
4.2.
4.3.
4.4.
4.5.
5.
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.9.
5.10.
5.11.
5.12.
5.13.
5.14.
6.11.
6.12.
6.13.
6.14.
6.15.
...
3.29
...
...
...
...
...
...
...
...
...
3.29
3.30
3.31
3.31
3.32
3.35
4.1–4.11
4.1
4.1
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
4.1
4.7
4.8
4.11
5.1–5.60
5.1
5.1
5.2
5.5
5.11
5.15
5.16
5.19
5.29
5.37
5.37
5.48
5.52
5.53
5.57
6.1–6.52
6.1
6.2
6.3
6.4
...
(v)
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6.16.
6.17.
6.18.
6.19.
6.20.
6.21
7.
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
7.8.
7.9.
7.10
7.11
7.12
7.12A
8.
8.1.
8.2.
8.3.
8.4.
8.5.
8.6.
8.7.
8.8.
8.9.
8.10.
8.11.
8.12.
8.13.
8.14.
8.15.
8.16.
8.17.
8.18.
8.19.
8.20.
8.21.
8.21.
9.
9.1.
9.2.
9.3.
9.4.
Second logarithmic ratio test
...
6.27
Kummer’s test
...
6.36
Gauss’s test
...
6.37
Cauchy’s integral test
..
6.41
Cauchy’s condensation test
...
6.47
An important auxiliary series
...
6.48
Objective questions
...
6.49
INFINITE SERIES WITH POSITIVE AND NEGATIVE TERMS ... 7.1–7.30
Introduction
...
7.1
Absolute convergence and conditional convergence
...
7.1
Alternating series
...
7.2
Cauchy principle of convergence for a series
...
7.10
Some important theorems on absolutely convergent series
...
7.11
Dirichlet’s theorem
...
7.12
Abel’s theorem
...
7.13
Re-arrangements of series
...
7.14
Re-arrangements of a conditionally convergent series
...
7.16
Objective questions
...
7.18
Modified forms of some important theorems
...
7.20
Additional solved examples
...
7.21
Cauchy product of two infinite series
...
7.25
Solved examples based on Art. 7.12
...
7.29
REAL FUNCTIONS. LIMIT AND CONTINUITY
... 8.1–8.55
Introduction
...
8.1
Algebraic operations on functions
...
8.2
Bounded and unbounded functions
...
8.2
Limit of a function
...
8.3
Algebra of limits
...
8.4
One-sided limits — right-hand and left-hand limits
...
8.8
Limits at infinity and infinite limits.
...
8.10
Characterization of the limit of a function at a point in terms of sequences ...
8.14
Cauchy’s criterion for finite limits
...
8.15
The four functional limits at a point
...
8.16
Continuous functions
...
8.16
Discontinutiy of a function
...
8.17
Algebra of continuous functions
...
8.26
Function of a function. Composite of functions
...
8.28
Continuity of the composite function
...
8.29
Criteria for continuity. Equivalent definition of continuity
...
8.29
Some properties of the continuity of a function at a point
...
8.35
Properties of functions continuous in closed finite intervals
...
8.37
Existence of the nth root of a given positive real number
...
8.42
Uniform continuity
...
8.43
Evaluation of lim (sin x ) / x, x being measured in radians
...
8.48
x o
Continuity of the inverses of continuous functions
...
8.49
Root function
...
8.50
Objective questions
...
8.51
REAL FUNCTIONS. THE DERIVATIVE
.... 9.1–9.21
Derivability of a function
...
9.1
A necessary condition for the existence of a finite derivative
...
9.2
Algebra of derivatives
...
9.3
Geometrical meaning of the derivative
...
9.6
(vi)
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9.5.
9.6.
Meaning of the sign of derivative at a point
Darboux’s theorem
...
Objective questions
...
10.
MEAN VALUE THEOREMS
...
10.1.
Rolle’s theorem
...
10.2.
Failure of Rolle’s theorem
...
10.3.
Geometrical interpretation of Rolle’s theorem
...
10.4.
Lagrange’s mean value theorem or first mean value theorem
...
10.5.
Increasing and decreasing functions. Monotonic functions
...
10.6.
Some useful deductions from the mean value theorem
...
10.7.
Increasing and decreasing functions and their application in establishing
some inequalities
...
10.8.
Cauchy’s mean value theorem
...
10.9.
Generalised mean value theorem
...
10.10. Higher derivatives
...
10.11. Taylor’s theorem with Schlomilch and Roche form of remainder
...
10.11A. Taylor’s theorem with Cauchy’s form of remainder
...
10.11B. Taylor’s theorem with Lagrange’s form of remainder
...
11.12. Power series representation of functions
...
10.13. Maclaurin’s infinite series
...
10.14. Some standard results
...
10.15. Powers series expansions of some standard functions
...
10.16. Vortex function
...
11.
MAXIMA AND MINIMA
...
11.1.
Introduction
...
11.2.
A necessary condition for the existence of extreme values
...
11.3.
Sufficient criteria for the existence of extreme values
...
11.4.
Applications to problems
...
Objective questions
...
12.
INDETERMINATE FORMS
12.1.
Introduction
...
12.2.
The indeterminate form (0/0)
..
L’Hopital’s theorem
...
12.3.
The indeterminate form (/)
...
12.4.
Some useful results
...
12.5.
Working rule to evaluate limit in indeterminate form (0/0)
...
12.6.
Application of L’ Hopital’s rule for / form
...
12.7.
The indeterminate form 0 × 
...
12.8.
The indeterminate form  – 
...
12.9.
The indeterminate forms 00, 0 and 1
...
Objective questions
...
13.
RIEMANN INTEGRABILITY
...
13.1
Introduction
...
13.2
Partitions and Riemann (or Darboux) sums
...
13.3
Some properties of Darboux sums
...
13.4.
Upper and lower Riemann integrals. Riemann integral
...
13.5
Another equivalent definition of integrability and integral
...
13.6
A second definition of Riemann integrability
...
Summation of series Theorem
...
13.7
Necessary and sufficient condition for integrability
...
(vii)
...
9.15
9.16
9.18
10.1–10.45
10.1
10.2
10.2
10.9
10.10
10.11
10.19
10.24
10.25
10.27
10.27
10.28
10.28
10.29
10.30
10.30
10.31
10.42
11.1–11.14
11.1
11.1
11.2
11.7
11.12
12.1–12.20
12.1
12.1
12.2
12.3
12.4
12.5
12.9
12.10
12.11
12.12
12.18
13.1–13.62
13.1
13.1
13.2
13.6
13.16
13.17
13.19
13.22
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13.8.
13.9
13.10
13.11
Particular classes of bounded integrable functions
Properties of integrable functions
Integrability of the sum, difference, product and quotient of
integrable functions
Integrability of the modulus of a bounded integrable function
...
...
13.25
13.29
...
...
13.30
13.33
13.12
Definition of
...
13.35
13.13
13.14
13.15
13.16
13.17
13.18
13.19
13.20
Inequalities for an integral
...
Functions defined by definite integrals
...
Fundamental theorem of integral calculus
...
Generalized mean value theorem
...
Abel's lemma.
...
Second mean value theorem
...
Change of variable in an integral
...
Integration by parts
...
Objective questions
...
THE RIEMANN STIELTJES INTEGRAL
...
Introduction
...
Partition of a set. Lower and upper Rieman–Stieltjes sums
...
The lower and upper Riemann–Stieltjes integrals
...
The Riemann–Stieltjes integrals
...
The Riemann–Stieltjes integral as a limit of sum
...
Some useful inequalities related to R-S integrals
...
Algebra of R-S integrable functions
...
Reduction of Riemann–Stielijes integral into Riemann integral
...
Some useful theorems
...
Objective questions
...
UNIFORM CONVERGENCE OF SEQUENCES
AND SERIES OF FUNCTIONS
...
Introduction
...
Cauchy’s general principle of uniform convergence
...
A test for uniform convergence of sequence of functions
...
Countinuity of the uniform limit of a uniformly convergent sequence
of continuous functions
...
Integrability of uniform limit of a uniformly convergent sequence of
integrable functions
...
Derivability of the point-wise limit of a sequence of derivable functions
if the derivatives are continuous and the sequence of derivatives is
uniformly convergent
...
Infinite series of functions
...
Test for the uniform convergence of a series
...
15.8.1. Cauchy’s general principle of convergence
...
15.8.2. Weierstrass’ M-Test for uniform convergence
...
Abel’s test and Dirichlet’s test
...
Proberties of uniformly convergent series of functions
...
The Weierstrass approximation theorem
...
Objective questions
...
b
14.
14.1
14.2
14.3
14.4.
14.5.
14.6.
14.7.
14.8.
14.9
15.
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10
15.11
 f(x) dx, if b  a
a
13.36
13.39
13.40
13.41
13.42
13.42
13.44
13.45
13.58
14.1–14.32
14.1
14.1
14.4
14.5
14.7
14.16
14.18
14.26
14.28
14.31
15.1–15.34
15.1
15.2
15.5
15.7
15.9
15.13
15.14
15.14
15.14
15.14
15.18
15.21
15.29
15.31
(viii)
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16.
16.1
16.2
16.3
16.4
IMPROPER INTEGRALS
Proper and improper integrals
Convergence of improper integrals of the first kind
Test for convergence at ‘a’. Positive integrand
The necessary and sufficient condition for the convergence of
the improper integral
16.5
Comparison of two integrals
16.5A A practical comparison test
16.6
Useful comparison integrals
16.7
Two useful tests
16.8
f(x) not necessarily positive. General test for convergence
16.9
Absolute and conditionally convergence
16.10
Convergence of improper integrals of the second kind
16.11
Convergence at . The integrand being positive
16.12
Comparison of two integrals
16.13
A useful comparison integral
16.14
General test. Convergence at . The integrand being not
necessarily positive. Cauchy’s test for convergence
16.15
Absolute and conditionally convergence of improper integrals of
second kind
16.16
Test for the absolute convergence of the integral of a product
16.17
Abel’s test
16.18
Dirichlet’s test
Objective questions
17.
POWER SERIES
17.1
Introduction
17.2
Power seriers
17.3
Some important facts about the power series
17.4
Radius of convergence and interval of convergence
17.5
Formulas for determining the radius of convergence
17.5A Solved examples based on Art. 17.5
17.6
Some basis theorems
17.7
Properties of functions expressible as power series
17.8
Abel’s theorem
17.9
Some theorems on power series
17.10
Solved examples
17.11
Elementary functions
18.
DOUBLE SEQUENCES AND SERIES
18.1
Double sequence
18.2
Convergence of a double sequence
18.3
Cauchy general principle of convergence of double sequence
18.4
Repeated double limits
18.5
Double series
18.6
Convergence of a double series
18.7
Double series of positive terms
18.8
Some tests for convergence of a double series of positive terms
18.9
Mixed series
18.10
Solved examples
18.11
Taylor’s theorem for power series
19.
METRIC SPACES
19.1
Introduction
19.2
Metric space
...
...
...
16.1–16.36
16.1
16.1
16.4
...
...
...
...
...
...
...
...
...
...
...
16.5
16.5
16.6
16.7
16.7
16.14
16.15
16.16
16.18
16.19
16.19
...
16.25
...
...
...
...
...
16.25
16.26
16.26
16.27
16.34
17.1–17. 20
17.1
17.1
17.1
17.2
17.2
17.3
17.5
17.6
17.9
17.12
17.14
17.17
18.1–18.12
18.1
18.1
18.1
18.1
18.2
18.4
18.5
18.6
18.7
18.8
18.11
19.1–19.76
19.1
19.1
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
(ix)
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19.3
19.4
19.5
19.6
19.7
19.8
19.9
19.10
19.11
19.12
19.13
19.14
19.15
19.16
19.17
19.18
19.19
19.20
19.21
Pseudo-metric space
...
19.1
Some important results for direct applications
...
19.1
Examples of metric spaces
...
19.2
Distance between two sets. Diameter of a set
...
19.4
Bounded and unbounded metric spaces
...
19.4
Open and closed spheres (or balls)
...
19.5
Neighbourhood of a point
...
19.5
Open set
...
19.5
Properties of open sets
...
19.6
Equivalent metrics
...
19.7
Limit Point (or accumulation point or a cluster point or a condensation point) 19.10
Closed set
...
19.10
Properties of closed sets
...
19.10
Subspaces
...
19.11
Adherent point (or contact point). Closure of a set
...
19.12
Properties of closure of a set
...
19.12
Interior point of a set. Interior of a set
...
19.13
Properties of interior of a set
...
19.14
Dense (or everywhere dense), dense in itself, nowhere
dense (or non-dense) and perfect sets
...
19.15
19.22
Separable space
...
19.15
19.23
Exterior, frontier and boundary points
...
19.16
19.24
Product of metric spaces
...
19.16
19.25
Continuous functions on metric spaces
...
19.17
19.26
Properties of continuous functions
...
19.17
19.27
Uniform continuity
...
19.18
19.28
Homeomorphism
...
19.19
19.29
Isometry
...
19.19
19.30
Sequence in a metric space
...
19.20
19.31
Cauchy sequence
...
19.23
19.32
Complete metric space
...
19.24
19.33
Properties of complete metric spaces
...
19.26
19.34
Cantor’s intersection theorem
...
19.27
19.35
Contraction mapping principle
...
19.30
Banach’s fixed point theorem
...
19.30
19.36
A subset of first category and second category
...
19.31
19.37
Compact metric space
...
19.32
Finite intersection property
...
19.34
Balzano-Weierstrass property (BWP)
...
19.34
Sequentially compact metric space
...
19.34
Countably compact metric space
...
19.34
19.37A  -net and total boundedness
...
19.35
19.37B Lebesgue number for a cover
...
19.36
Lebesgue covering lemma
...
19.36
19.37C Locally compact metric space
...
19.38
19.38
Connected metric spaces
...
19.40
Separated sets
...
19.40
Connected and disconnected sets
...
19.40
19.39
Components of a metric spaces
...
19.46
19.40
Connectedness of product of connected metric spaces
...
19.47
Objective questions
...
19.47
Additional problem on Chapter 19
...
19.53
20.
BETA AND GAMMA FUNCTIONS
20.1–20.26
20.1
Introduction
...
20.1
20.2
Euler’s integrals. Beta and Gamma functions
...
20.1
(x)
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20.3
20.4
Properties of Gamma function
Extension of definiton of Gamma function
...
...
20.5
20.6
20.7
20.8
20.9
20.10
20.11
20.12
20.13
20.14
Theorem. To show that  (1/ 2)  
...
Transformation of Gamma function
...
Solved examples based on Gamma function
...
Symmetrical properties of Beta function
...
Evaluation of B (m, n) in an explicit form when m or n is a positive integer
Transformation of Beta function
...
Relation between Beta and Gamma functions
...
Solved examples
...
Legendre-duplication formula
...
Solved examples
...
Miscellaneous problems based on this Chapter
...
20.3
20.3
20.4
20.8
20.8
20.9
20.12
20.15
20.21
20.22
20.23
21.
21.1
21.2
21.3
DIFFERENTIATION UNDER THE INTEGRAL SIGN
...
Introduction
...
Leibnitz’s rule for differentiation under the integral sign
...
General form of Leibnitz’s rule of differentiation under the integral
sign when the limits of integration are functions of the prarameters ...
21.2
Evaluation of integral  f ( x,  )dx, where a and b are
21.4B
independent of  Working rule
Solved examples of type I based on Art 21.4A
21.5A
Evaluation of integral  g ( x,  , ) dx, where a and b are
21.5B
independent of parameters  and  Working rule
Solved examples of type 2 based on Art. 21.5 A
21.6A
Evaluation of integral  g ( ) f ( x,  ) dx, where g() and h() are
21.7B
21.1–21.24
21.1
21.1
b
21.4A
21.6B
21.7A
20.1
20.2
a
...
...
21.4
21.4
...
...
21.14
21.14
functions of parameter . Working rule
Solved examples of types 3 based on Art. 21.6 A
Determination of the value of an integral when certain standard
known integral is given with its value. Working rule
Solved examples of type 4 based on Art 21.7A
...
...
21.19
21.19
...
...
21.21
21.21
INDEX
...
I.1-I.4
b
a
h ( )
Dedicated to memory
of my parents
— M.D. Raisinghania
(xi)
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SYMBOLS
the set of natural numbers
the set of integers
the set of positive integers
the set of rational numbers
the set of positive rational numbers
the set of real numbers
the set of positive real numbers
N
Z or
I
+
Z or I+
Q
Q+
R
R+

implies

is equivalent to
{}
set

:
is an element of
such that
is contained in (is a subset of)
contains (is a superset of)


A or ~A or A or U ~ A or U\A


complement of A with respect to U
complement of A with respect to R
union
intersection

the empty set

there exists

for all
c
A or ~A or Ac or R ~ A or R\A
THE GREEK ALPHABET
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
nu





A
B


E


Z







I
K

M
N
H
xi
omicron
pi
rho
sigma
tau
upsilon
phi
chi
psi
omega

o



O

P








T


X


(xii)
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Chapter
1
Sets and Functions
1.1. INTRODUCTION
Set-Theoretic Notation and Terminology. What is called Real Analysis is a development
of the set of real numbers which is reached through a series of successive extensions and
generalisations starting from the set of natural numbers. As a matter of fact, starting from the
set of natural numbers, we first pass on to the set of integers, then to the set of rational numbers
and finally to the set of real numbers. Of course, from the set of real numbers, we can also pass
on to the set of complex numbers but since Real Analysis is not concerned with complex
numbers, we shall have nothing to do with complex numbers in this course. What is known as
Complex Analysis is a development of the set of complex numbers.
It is no part of the plan of this book to construct the various systems of numbers and to
develop their properties axiomatically from a given system of postulates. All that is intended
here is to describe the various properties of these systems and to bring out the essential
differences between them; it being understood that most of these properties are already familiar
to the reader. What is important from the point of view of this course is the form in which these
properties are stated and the type of emphasis which is thus brought out. While this programme
will be undertaken in the following Chapter 2, we propose in this chapter to introduce some
new notations and terms pertaining to ‘sets’ and ‘functions’ which will be found useful for the
exposition of the subject proposed to be studied in this book.
1.2. STATEMENTS
In our everyday language, we are concerned with statements which are often distinguished
as Interrogative, Imperative, Exclamatory or Declarative. In Mathematics, however, our chief
interest is only in those statements which are Declarative and which may be either true or false.
Consider, for example, the following statements, some of which are true and some false :
(i) The sum of the three angles of a triangle is equal to two right angles.
(ii) Every rectangle is a triangle.
(iii) The sum of an opposite pair of angles of a cyclic quadrilateral is equal to two right
angles.
(iv) If two straight lines are perpendicular to the same straight line, then they are parallel
to each other.
(v) The straight line joining the mid-points of two sides of a triangle is parallel to the third.
(vi) If x is 2 and y is 5, then x  y is 7.
(vii) If xy  0 and x, y are real numbers, then x  0 and y  0.
(viii) If xy  1 and x, y are natural numbers, then x  1 and y  1.
(ix) If xy  1 and x, y are rational numbers, then x  1 and y  1.
(x) If x  3, then x2  9.
1.1
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1.2
Elements of Real Analysis
(xi) If x2  9, then x  3.
(xii) Every pair of natural numbers admits of a highest common factor.
(xiii) Every natural number admits of an infinite number of factors.
1.3. CONNECTIVES : , , , , 
In the study of Mathematics, we are concerned with logical inter-connections between statements.
Also on the basis of given statements, we build up new statements. There are a few symbols
which are found useful for describing logical inter-connections between given statements and for
building up new statements from the old ones and we now proceed to introduce these symbols.
The Connective  : If P and Q be two statements such that the truth of the statement P
implies the truth of the statement Q, we exhibit this relationship between the two statements
symbolically as P  Q so that the symbol, , stands for ‘Implies’. Thus, we have
(i) x  3x2  9.
(ii) x is a real number  x2  0.
(iii) ABCD is a parallelogram  AB  CD. (iv) ABC is a triangle  AB  BC  AC.
(v) AB CD and CD EF  AB EF.
The statement P  Q formulates what is often expressed in any one of the following ways :
(i) If P then Q.
(ii) A necessary condition for the truth of P is the truth of Q.
(iii) A sufficient condition for the truth of Q is the truth of P.
The symbol  is a connective in as much as connecting the two statements P, Q, it generates
a third statement, viz., the statement P  Q.
The Connective  : If P, Q are two statements such that we have P  Q as well as
Q  P, we write P  Q and say that P implies and is implied by Q or that P is equivalent
to Q.
Thus, ABCD is a parallelogram  AB  CD and BC  AD.
x2  9, x, y are real numbers  x  {3, – 3}.
The statement P  Q expresses what is also described in one of the following ways :
(i) P if and only if Q.
(ii) Q if and only if P.
(iii) A necessary and sufficient condition for the truth of P is the truth of Q.
(iv) A necessary and sufficient condition for the truth of Q is the truth of P.
(v) P and Q are equivalent statements.
The Connectives ,  : The symbols , , stand for and, or respectively.
If P and Q be two statements, then with the help of these connectives, we form two new
statements P  Q, P  Q.
The statement P  Q is true if and only if P is true as well as Q is true. Thus, if the
statements P, Q are both true, then the statement P  Q is true and vice-versa.
The statement P  Q is true if and only if P is true or Q is true, i.e., if and only if at least
one of P and Q is true.
For example, ABCD is a parallelogram  AB  CD  BC  AD.
x2 – 5x  6  0  x  2  x  3.
xy  0, x, y are real numbers  x  0  y  0.
x  2  y  3  x  y  5.
Negation  : If P denotes a statement, then  P, denotes the negation or the denial of P.
Let P denote the statement x  4. Then,  P, denotes the statement x  4.
For example, ABC is a triangle   [AB  BC  AC].
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