Real Variables with Basic Metric Space Topology

INTRODUCTION Basic Terminology Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets Distance and Convergence Minicourse in Basic Logic Limit Points and Closure Review Problems for Chapter 1 SOME BASIC TOPOLOGICAL PROPERTIES OF Rp Unions and Intersections of Open and Closed Sets Compactness Some Applications of Compactness Least Upper Bounds and Completeness Review Problems for Chapter 2 UPPER AND LOWER LIMITS OF SEQUENCES OF REAL NUMBERS Generalization of the Limit Concept Some Properties of Upper and Lower Limits Convergence of Power Series Review Problems for Chapter 3 CONTINUOUS FUNCTIONS Continuity: Ideas, Basic Terminology, Properties Continuity and Compactness Types of Discontinuities The Cantor Set Review Problems for Chapter 4 DIFFERENTIATION The Derivative and Its Basic Properties Additional Properties of the Derivative; Some Applications of the Mean Value Theorem Review Problems for Chapter 5 RIEMANN-STIELTJES INTEGRATION Definition of the Integral Properties of the Integral Functions of Bounded Variation Some Useful Integration Theorems Review Problems for Chapter 6 UNIFORM CONVERGENCE AND APPLICATIONS Pointwise and Uniform Convergence Uniform Convergence and Limit Operations The Weierstrass M-test and Applications Equicontinuity and the Arzela-Ascoli Theorem The Weierstrass Approximation Theorem Review Problems for Chapter 7 FURTHER TOPOLOGICAL RESULTS The Extension Problem Baire Category Theorem Connectedness Semicontinuous Functions Review Problems for Chapter 8 EPILOGUE Some Compactness Results Replacing Cantor's Nested Set Property The Real Numbers Revisited SOLUTIONS TO PROBLEMS INDEX