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.

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.

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.