450 Pages
60 B/W Illustrations
by
Chapman & Hall
436 Pages
by
Chapman & Hall
Also available as eBook on:
Focusing on one of the main pillars of mathematics, Elements of Real Analysis provides a solid foundation in analysis, stressing the importance of two elements. The first building block comprises analytical skills and structures needed for handling the basic notions of limits and continuity in a simple concrete setting while the second component involves conducting analysis in higher dimensions... Read more
PREFACE
PRELIMINARIES
Sets
Functions
REAL NUMBERS
Field Axioms
Order Axioms
Natural Numbers, Integers, Rational Numbers
Completeness Axiom
Decimal Representation of Real Numbers
Countable Sets
SEQUENCES
Sequences and Convergence
Properties of Convergent Sequences
Monotonic Sequences
The Cauchy Criterion
Subsequences
Upper and Lower Limits
Open and Closed Sets
INFINITE SERIES
Basic Properties
Convergence Tests
LIMIT OF A FUNCTION
Limit of a Function
Basic Theorems
Some Extensions of the Limit
Monotonic Functions
CONTINUITY
Continuous Functions
Combinations of Continuous Functions
Continuity on an Interval
UniformContinuity
Compact Sets and Continuity
DIFFERENTIATION
The Derivative
TheMean Value Theorem
L'Hôpital's Rule
Taylor's Theorem
THE RIEMANN INTEGRAL
Riemann Integrability
Darboux's Theorem and Riemann Sums
Properties of the Integral
The Fundamental Theorem of Calculus
Improper Integrals
SEQUENCES AND SERIES OF FUNCTIONS
Sequences of Functions
Properties of Uniform Convergence
Series of Functions
Power Series
LEBESGUE MEASURE
Classes of Subsets of R
Lebesgue Outer Measure
Lebesgue Measure
Measurable Functions
LEBESGUE INTEGRATION
Definition of the Lebesgue Integral
Properties of the Lebesgue Integral
Lebesgue Integral and Pointwise Convergence
Lebesgue and Riemann Integrals
REFERENCES
NOTATION
INDEX
PRELIMINARIES
Sets
Functions
REAL NUMBERS
Field Axioms
Order Axioms
Natural Numbers, Integers, Rational Numbers
Completeness Axiom
Decimal Representation of Real Numbers
Countable Sets
SEQUENCES
Sequences and Convergence
Properties of Convergent Sequences
Monotonic Sequences
The Cauchy Criterion
Subsequences
Upper and Lower Limits
Open and Closed Sets
INFINITE SERIES
Basic Properties
Convergence Tests
LIMIT OF A FUNCTION
Limit of a Function
Basic Theorems
Some Extensions of the Limit
Monotonic Functions
CONTINUITY
Continuous Functions
Combinations of Continuous Functions
Continuity on an Interval
UniformContinuity
Compact Sets and Continuity
DIFFERENTIATION
The Derivative
TheMean Value Theorem
L'Hôpital's Rule
Taylor's Theorem
THE RIEMANN INTEGRAL
Riemann Integrability
Darboux's Theorem and Riemann Sums
Properties of the Integral
The Fundamental Theorem of Calculus
Improper Integrals
SEQUENCES AND SERIES OF FUNCTIONS
Sequences of Functions
Properties of Uniform Convergence
Series of Functions
Power Series
LEBESGUE MEASURE
Classes of Subsets of R
Lebesgue Outer Measure
Lebesgue Measure
Measurable Functions
LEBESGUE INTEGRATION
Definition of the Lebesgue Integral
Properties of the Lebesgue Integral
Lebesgue Integral and Pointwise Convergence
Lebesgue and Riemann Integrals
REFERENCES
NOTATION
INDEX
Biography
M.A. Al-Gwaiz, S.A. Elsanousi
