A Guide to Real Variables /

A Guide to Real Variables provides aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness,...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Krantz, Steven G. (Steven George), 1951-
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, 2012.
Colección:Dolciani mathematical expositions.
Temas:
Functions of real variables.
Fonctions de variables réelles.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
Functions of real variables
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Contents
  • Preface
  • 1 Basics
  • 1.1 Sets
  • 1.2 Operations on Sets
  • 1.3 Functions
  • 1.4 Operations on Functions
  • 1.5 Number Systems
  • 1.5.1 The Real Numbers
  • 1.6 Countable and Uncountable Sets
  • 2 Sequences
  • 2.1 Introduction to Sequences
  • 2.1.1 The Definition and Convergence
  • 2.1.2 The Cauchy Criterion
  • 2.1.3 Monotonicity
  • 2.1.4 The Pinching Principle
  • 2.1.5 Subsequences
  • 2.1.6 The Bolzano-Weierstrass Theorem
  • 2.2 Limsup and Liminf
  • 2.3 Some Special Sequences
  • 3 Series
  • 3.1 Introduction to Series
  • 3.1.1 The Definition and Convergence3.1.2 Partial Sums
  • 3.2 Elementary Convergence Tests
  • 3.2.1 The Comparison Test
  • 3.2.2 The Cauchy Condensation Test
  • 3.2.3 Geometric Series
  • 3.2.4 The Root Test
  • 3.2.5 The Ratio Test
  • 3.2.6 Root and Ratio Tests for Divergence
  • 3.3 Advanced Convergence Tests
  • 3.3.1 Summation by Parts
  • 3.3.2 Abelâ€?s Test
  • 3.3.3 Absolute and Conditional Convergence
  • 3.3.4 Rearrangements of Series
  • 3.4 Some Particular Series
  • 3.4.1 The Series for e
  • 3.4.2 Other Representations for e
  • 3.4.3 Sums of Powers
  • 3.5 Operations on Series3.5.1 Sums and Scalar Products of Series
  • 3.5.2 Products of Series
  • 3.5.3 The Cauchy Product
  • 4 The Topology of the Real Line
  • 4.1 Open and Closed Sets
  • 4.1.1 Open Sets
  • 4.1.2 Closed Sets
  • 4.1.3 Characterization of Open and Closed Sets in Terms of Sequences
  • 4.1.4 Further Properties of Open and Closed Sets
  • 4.2 Other Distinguished Points
  • 4.2.1 Interior Points and Isolated Points
  • 4.2.2 Accumulation Points
  • 4.3 Bounded Sets
  • 4.4 Compact Sets
  • 4.4.1 Introduction
  • 4.4.2 The Heine-Borel Theorem
  • 4.4.3 The Topological Characterization of Compactness4.5 The Cantor Set
  • 4.6 Connected and Disconnected Sets
  • 4.6.1 Connectivity
  • 4.7 Perfect Sets
  • 5 Limits and the Continuity of Functions
  • 5.1 Definitions and Basic Properties
  • 5.1.1 Limits
  • 5.1.2 A Limit that Does Not Exist
  • 5.1.3 Uniqueness of Limits
  • 5.1.4 Properties of Limits
  • 5.1.5 Characterization of Limits Using Sequences
  • 5.2 Continuous Functions
  • 5.2.1 Continuity at a Point
  • 5.2.2 The Topological Approach to Continuity
  • 5.3 Topological Properties and Continuity
  • 5.3.1 The Image of a Function5.3.2 Uniform Continuity
  • 5.3.3 Continuity and Connectedness
  • 5.3.4 The Intermediate Value Property
  • 5.4 Monotonicity and Classifying Discontinuities
  • 5.4.1 Left and Right Limits
  • 5.4.2 Types of Discontinuities
  • 5.4.3 Monotonic Functions
  • 6 The Derivative
  • 6.1 The Concept of Derivative
  • 6.1.1 The Definition
  • 6.1.2 Properties of the Derivative
  • 6.1.3 The Weierstrass Nowhere Differentiable Function
  • 6.1.4 The Chain Rule
  • 6.2 The Mean Value Theorem and Applications
  • 6.2.1 Local Maxima and Minima

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