Cover -- Preface -- Contents -- Chapter 1 Real Numbers -- 1.1 Introduction -- 1.2 Field Structure and Order Structure -- 1.3 Bounded and Unbounded Sets: Supremum, Infimum -- 1.4 Completeness in the Set of Real Numbers -- 1.5 Absolute Value of a Real Number -- Chapter 2 Limit Points: Open and Closed Sets -- 2.1 Introduction -- 2.2 Limit Points of a Set -- 2.3 Closed Sets: Closure of a Set -- Chapter 3 Real Sequences -- 3.1 Functions -- 3.2 Sequences -- 3.3 Limit Points of a Sequence -- 3.4 Convergent Sequences -- 3.5 Non-Convergent Sequences (Definitions) -- 3.6 Cauchy's General Principle of Convergence -- 3.7 Algebra of Sequences -- 3.8 Some Important Theorems -- 3.9 Monotonic Sequences -- Chapter 4 Infinite Series -- 4.1 Introduction -- 4.2 Positive Term Series -- 4.3 Comparison Tests for Positive Term Series -- 4.4 Cauchy's Root Test -- 4.5 D'Alembert's Ratio Test -- 4.6 Raabe's Test -- 4.7 Logarithmic Test -- 4.8 Integral Test -- 4.9 Gauss's Test -- 4.10 Series with Arbitrary Terms -- Chapter 5 Functions with Interval as Domain (I) -- 5.1 Limits -- 5.2 Continuous Functions -- 5.3 Functions Continuous on Closed Intervals -- 5.4 Uniform Continuity -- Chapter 6 Functions with Interval as Domain (II) -- 6.1 The Derivative -- 6.2 Continuous Functions -- 6.3 Increasing and Decreasing Functions -- 6.4 Darboux's Theorem -- 6.5 Rolle's Theorem -- 6.6 Lagrange's Mean Value Theorem -- 6.7 Cauchy's Mean Value Theorem -- 6.8 Higher Order Derivatives -- Chapter 7 Applications of Taylor's Theorem -- 7.1 Extreme Values (Definitions) -- 7.2 Indeterminate Forms -- Chapter 8 Elementary Functions -- 8.1 Introduction -- 8.2 Power Series -- 8.3 Exponential Functions -- 8.4 Logarithmic Functions (base e) -- 8.5 Trigonometric Functions -- Chapter 9 The Riemann Integral -- 9.1 Introduction -- 9.2 Definitions and Existence of the Integral.

9.3 Refinement of Partitions -- 9.4 Darboux's Theorem -- 9.5 Conditions of Integrability -- 9.6 Integrability of the Sum and Difference of Integrable Functions -- 9.7 The Integral as a Limit of Sums (Riemann Sums) -- 9.8 Some Integrable Functions -- 9.9 Integration and Differentiation (The Primitive) -- 9.10 The Fundamental Theorem of Calculus -- 9.11 Mean Value Theorems of Integral Calculus -- 9.12 Integration By Parts -- 9.13 Change of Variable in an Integral -- 9.14 Second Mean Value Theorem -- Chapter 10 The Riemann-Stieltjes Integral -- 10.1 Definitions and Existence of the Integral -- 10.2 A Condition of Integrability -- 10.3 Some Theorems -- 10.4 A Definition (Integral as a Limit of Sum) -- 10.5 Some Important Theorems -- Chapter 11 Functions of Several Variables -- 11.1 Explicit and Implicit Functions -- 11.2 Continuity -- 11.3 Partial Derivatives -- 11.4 Differentiability -- 11.5 Partial Derivatives of Higher Order -- 11.6 Differentials of Higher Order -- 11.7 Functions of Functions -- 11.8 Change of Variables -- 11.9 Taylor's Theorem -- 11.10 Extreme Values: Maxima and Minima -- 11.11 Functions of Several Variables -- Chapter 12 Implicit Functions -- 12.1 Definition -- 12.2 Jacobians -- 12.3 Stationary Values Under Subsidiary Conditions -- Appendix I-Theorems on Rearrangement of Terms and Tests for Arbitrary Series -- 1. Tests for Arbitrary Term Series -- 2. Rearrangement of Terms -- Appendix II-Cantor's Theory of Real Numbers -- 1. Introduction -- 2. Sequences of Rational Numbers -- 3. Real Numbers -- 4. Addition and Multiplication in R -- 5. Order in R -- 6. Real Rational and Irrational Numbers -- 7. Some Properties of Real Numbers -- 8. Completeness in R -- Bibliography -- Index.