FUNDAMENTAL REAL ANALYSIS
CONTENTS CHAPTER THE REAL NUMBERS 1 1 THE REAL NUMBER SYSTEM 1 2 MATHEMATICAL INDUCTION 1 3 THE REAL LINE CHAPTER DI?ERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE 2 1 FUNCTIONS AND LIMITS 2 2 CONTINUITY 2 3 DI?ERENTIABLE FUNCTIONS OF ONE VARIABLE 2 4 L’HOSPITAL’S RULE 2 5 TAYLOR’S THEOREM INTEGRAL CALCULUS OF FUNCTIONS OF ONE VARIABLE 3 1 DE?NITION OF THE INTEGRAL 3 2 EXISTENCE OF THE INTEGRAL 3 3 PROPERTIES OF THE INTEGRAL 3 4 IMPROPER INTEGRALS 3 5 A MORE ADVANCED LOOK AT THE EXISTENCE OF THE PROPER RIEMANN INTEGRAL IN?NITE SEQUENCES AND SERIES 4 1 SEQUENCES OF REAL NUMBERS 4 2 EARLIER TOPICS REVISITED WITH SEQUENCES 4 3 IN?NITE SERIES OF CONSTANTS 4 4 SEQUENCES AND SERIES OF FUNCTIONS 4 5 POWER SERIES REAL-VALUED FUNCTIONS OF SEVERAL VARIABLES 5 1 STRUCTURE OF RN 5 2 CONTINUOUS REAL-VALUED FUNCTION OF N VARIABLES 5 3 PARTIAL DERIVATIVES AND THE DI?ERENTIAL 5 4 THE CHAIN RULE AND TAYLOR’S THEOREM VECTOR-VALUED FUNCTIONS OF SEVERAL VARIABLES 6 1 LINEAR TRANSFORMATIONS AND MATRICES 6 2 CONTINUITY AND DI?ERENTIABILITY OF TRANSFORMATIONS 6 3 THE INVERSE FUNCTION THEOREM 6 4 THE IMPLICIT FUNCTION THEOREM INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES 7 1 DE?NITION AND EXISTENCE OF THE MULTIPLE INTEGRAL 7 2 ITERATED INTEGRALS AND MULTIPLE INTEGRALS 7 3 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS METRIC SPACES 8 1 INTRODUCTION TO METRIC SPACES 8 2 COMPACT SETS IN A METRIC SPACE 8 3 CONTINUOUS FUNCTIONS ON METRIC SPACES ANSWERS TO SELECTED EXERCISES