calculus document

May 11th, 2015
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A ProblemText in Advanced CalculusJohn M. ErdmanPortland State UniversityVersion July 1, 2014c2005John M. ErdmanE-mail address: erdman@pdx.edu.iiTo ArgentinaContentsPREFACExiFOR STUDENTS: HOW TO USE THIS PROBLEMTEXTxvChapter 1. INTERVALS1.1. DISTANCE AND NEIGHBORHOODS1.2. INTERIOR OF A SET112Chapter 2. TOPOLOGY OF THE REAL LINE2.1. OPEN SUBSETS OF R2.2. CLOSED SUBSETS OF R557Chapter 3. CONTINUOUS FUNCTIONS FROM R TO R3.1. CONTINUITYAS A LOCAL PROPERTY3.2. CONTINUITYAS A GLOBAL PROPERTY3.3. FUNCTIONS DEFINED ON SUBSETS OF R991013Chapter 4. SEQUENCES OF REAL NUMBERS4.1. CONVERGENCE OF SEQUENCES4.2. ALGEBRAIC COMBINATIONS OF SEQUENCES4.3. SUFFICIENT CONDITION FOR CONVERGENCE4.4. SUBSEQUENCES1717192023Chapter 5. CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM5.1. CONNECTED SUBSETS OF R5.2. CONTINUOUS IMAGES OF CONNECTED SETS5.3. HOMEOMORPHISMS27272930Chapter 6. COMPACTNESS AND THE EXTREME VALUE THEOREM6.1. COMPACTNESS6.2. EXAMPLES OF COMPACT SUBSETS OF R6.3. THE EXTREME VALUE THEOREM33333436Chapter 7. LIMITS OF REAL VALUED FUNCTIONS7.1. DEFINITION7.2. CONTINUITY AND LIMITS393940Chapter 8. DIFFERENTIATION OF REAL VALUED FUNCTIONS8.1. THE FAMILIES O AND o8.2. TANGENCY8.3. LINEAR APPROXIMATION8.4. DIFFERENTIABILITY4343454647Chapter 9. METRIC SPACES9.1. DEFINITIONS9.2. EXAMPLES515152vviCONTENTS9.3.Chapter10.1.10.2.10.3.STRONGLY EQUIVALENT METRICS55

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