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050  4 QA331.5 -- .K68 2009eb 
082 0  515.8 
100 1  Krantz, Steven G 
245 10 Guide to Real Variables 
264  1 Washington :|bAmerican Mathematical Society,|c2009 
264  4 |c©2009 
300    1 online resource (164 pages) 
336    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
490 1  Dolciani Mathematical Expositions ;|vv.38 
505 0  Intro -- 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 Convergence -- 3.1.2 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 Abel'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 Series -- 3.5.1 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 Compactness -- 4.5 The Cantor Set -- 
       4.6 Connected and Disconnected Sets -- 4.6.1 Connectivity 
       -- 4.7 Perfect Sets -- 5  Limits and the Continuity of 
       Functions 
505 8  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 Function -- 5.3.2 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 -- 6.2.2 
       Fermat's Test -- 6.2.3 Darboux's Theorem -- 6.2.4 The Mean
       Value Theorem -- 6.2.5 Examples of the Mean Value Theorem 
       -- 6.3 Further Results on the Theory of Differentiation --
       6.3.1 l'Hopital's Rule -- 6.3.2 Derivative of an Inverse 
       Function -- 6.3.3 Higher Derivatives -- 6.3.4 Continuous 
       Differentiability -- 7  The Integral -- 7.1 The Concept of
       Integral -- 7.1.1 Partitions -- 7.1.2 Refinements of 
       Partitions -- 7.1.3 Existence of the Riemann Integral -- 
       7.1.4 Integrability of Continuous Functions -- 7.2 
       Properties of the Riemann Integral -- 7.2.1 Existence 
       Theorems -- 7.2.2 Inequalities for Integrals -- 7.2.3 
       Preservation of Integrable Functions Under Composition -- 
       7.2.4 The Fundamental Theorem of Calculus -- 7.2.5 Mean 
       Value Theorems -- 7.3 Further Results on the Riemann 
       Integral -- 7.3.1 The Riemann-Stieltjes Integral -- 7.3.2 
       Riemann's Lemma -- 7.4 Advanced Results on Integration 
       Theory 
505 8  7.4.1 Existence for the Riemann-Stieltjes Integral -- 
       7.4.2 Integration by Parts -- 7.4.3 Linearity Properties -
       - 7.4.4 Bounded Variation -- 8  Sequences and Series of 
       Functions -- 8.1 Partial Sums and Pointwise Convergence --
       8.1.1 Sequences of Functions -- 8.1.2 Uniform Convergence 
       -- 8.2 More on Uniform Convergence -- 8.2.1 Commutation of
       Limits -- 8.2.2 The Uniform Cauchy Condition -- 8.2.3 
       Limits of Derivatives -- 8.3 Series of Functions -- 8.3.1 
       Series and Partial Sums -- 8.3.2 Uniform Convergence of a 
       Series -- 8.3.3 The Weierstrass M-Test -- 8.4 The 
       Weierstrass Approximation Theorem -- 8.4.1 Weierstrass's 
       Main Result -- 9  Advanced Topics -- 9.1 Metric Spaces -- 
       9.1.1 The Concept of a Metric -- 9.1.2 Examples of Metric 
       Spaces -- 9.1.3 Convergence in a Metric Space -- 9.1.4 The
       Cauchy Criterion -- 9.1.5 Completeness -- 9.1.6 Isolated 
       Points -- 9.2 Topology in a Metric Space -- 9.2.1 Balls in
       a Metric Space -- 9.2.2 Accumulation Points -- 9.2.3 
       Compactness -- 9.3 The Baire Category Theorem -- 9.3.1 
       Density -- 9.3.2 Closure -- 9.3.3 Baire's Theorem -- 9.4 
       The Ascoli-Arzela Theorem -- 9.4.1 Equicontinuity -- 9.4.2
       Equiboundedness -- 9.4.3 The Ascoli-Arzela Theorem -- 
       Glossary of Terms from Real Variable Theory -- 
       Bibliography -- Index -- About the Author 
588    Description based on publisher supplied metadata and other
       sources 
590    Electronic reproduction. Ann Arbor, Michigan : ProQuest 
       Ebook Central, 2020. Available via World Wide Web. Access 
       may be limited to ProQuest Ebook Central affiliated 
       libraries 
650  0 Functions of real variables 
655  4 Electronic books 
776 08 |iPrint version:|aKrantz, Steven G.|tGuide to Real 
       Variables|dWashington : American Mathematical Society,
       c2009|z9780883853443 
830  0 Dolciani Mathematical Expositions 
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