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ABOUT OLYMPIADS
WHAT ARE THE OLYMPIADS ?
International Olympiads in Science and Mathematics are the world championships
for school students held every year in a different country. The most famous
Olympiads are The International Mathematical Olympiad (IMO), The International
Physics Olympiad (IPhO), The International Chemistry Olympiad (IChO), The
International Biology Olympiad (IBO), The International Olympiad in Informatics
(IOI) and The International Astronomy Olympiad (IAO). Countries send a
delegation of students and leaders to participate at the International level.
India conducts National Olympiads to identify the best students who can
represent India at the international level.
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 examination
S. CHAND & COMPANY LTD.
(AN ISO 9001: 2008 COMPANY)
RAM NAGAR, NEW DELHI-110 055 1. SETS AND FUNCTIONS ... 1.1–1.9
1.1. Introduction ... 1.1
1.2. Statements ... 1.1
1.3. Connectives ... 1.2
1.4. Sets ... 1.4
1.5. Functions (or mappings) ... 1.6
1.6. Composite of functions (or product of functions) ... 1.7
1.7. Inverse function ... 1.8
1.8. Binary operation ... 1.8
Objective questions ... 1.9
2. THE REAL NUMBERS ... 2.1–2.32
2.1. Introduction ... 2.1
2.2. The set N of natural numbers ... 2.2
2.3. The set I or Z of integers ... 2.3
2.4. The set Q of rational numbers ... 2.4
2.5. The set of real numbers R as a complete ordered field ... 2.6
2.6. Closed, open, semi-closed and semi-open intervals ... 2.7
2.7. 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 ... 2.7
2.8. Order-completeness of the set of real numbers ... 2.16
2.9. Equivalent descriptions of the order-completeness. Property of the
set of real numbers ... 2.18
2.10. Explicit statement of the properties of the set of real numbers as a
complete ordered field ... 2.20
2.11. Some important properties of the system of real numbers ... 2.20
2.12. The denseness property of the set of real numbers R ... 2.21
2.13. The modulus (or absolute value) of a real number ... 2.26
2.14. Arithmetic and geometric continua ... 2.29
Objective questions ... 2.31
3. NEIGHBOURHOODS AND LIMIT POINTS OF A SET.
OPEN AND CLOSED SETS ... 3.1–3.36
3.1. Introduction ... 3.1
3.2. Neighbourhood of a point ... 3.1
3.3. Properties of neighbourhoods ... 3.3
3.4. Limit (or accumulation or condensation) point of a set ... 3.4
3.5. Existence of limit points ... 3.8
3.6. Open and closed sets ... 3.13
3.7. Basic theorems concerning families of open and closed sets ... 3.15
3.8. Illustrations of open sets ... 3.18
3.9. Illustrations of closed sets ... 3.19
3.10. Interior point and interior of a set ... 3.22
3.11. Exterior point and exterior of a set ... 3.22
3.12. Boundary (or frontier) point and boundary (or frontier) of a set ... 3.22
3.13. Theorems on interior of a set ... 3.23
3.14. Adherent point (or a contact point) and closure of a set ... 3.25
3.15. Theorems on closure of a set ... 3.26
(iv)
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301. SETS AND FUNCTIONS ... 1.1–1.9
1.1. Introduction ... 1.1
1.2. Statements ... 1.1
1.3. Connectives ... 1.2
1.4. Sets ... 1.4
1.5. Functions (or mappings) ... 1.6
1.6. Composite of functions (or product of functions) ... 1.7
1.7. Inverse function ... 1.8
1.8. Binary operation ... 1.8
Objective questions ... 1.9
2. THE REAL NUMBERS ... 2.1–2.32
2.1. Introduction ... 2.1
2.2. The set N of natural numbers ... 2.2
2.3. The set I or Z of integers ... 2.3
2.4. The set Q of rational numbers ... 2.4
2.5. The set of real numbers R as a complete ordered field ... 2.6
2.6. Closed, open, semi-closed and semi-open intervals ... 2.7
2.7. 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 ... 2.7
2.8. Order-completeness of the set of real numbers ... 2.16
2.9. Equivalent descriptions of the order-completeness. Property of the
set of real numbers ... 2.18
2.10. Explicit statement of the properties of the set of real numbers as a
complete ordered field ... 2.20
2.11. Some important properties of the system of real numbers ... 2.20
2.12. The denseness property of the set of real numbers R ... 2.21
2.13. The modulus (or absolute value) of a real number ... 2.26
2.14. Arithmetic and geometric continua ... 2.29
Objective questions ... 2.31
3. NEIGHBOURHOODS AND LIMIT POINTS OF A SET.
OPEN AND CLOSED SETS ... 3.1–3.36
3.1. Introduction ... 3.1
3.2. Neighbourhood of a point ... 3.1
3.3. Properties of neighbourhoods ... 3.3
3.4. Limit (or accumulation or condensation) point of a set ... 3.4
3.5. Existence of limit points ... 3.8
3.6. Open and closed sets ... 3.13
3.7. Basic theorems concerning families of open and closed sets ... 3.15
3.8. Illustrations of open sets ... 3.18
3.9. Illustrations of closed sets ... 3.19
3.10. Interior point and interior of a set ... 3.22
3.11. Exterior point and exterior of a set ... 3.22
3.12. Boundary (or frontier) point and boundary (or frontier) of a set ... 3.22
3.13. Theorems on interior of a set ... 3.23
3.14. Adherent point (or a contact point) and closure of a set ... 3.25
3.15. Theorems on closure of a set ... 3.26
(iv)
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