A problem text in advanced calculus

A ProblemText in Advanced Calculus......Page 1
Contents......Page 5
PREFACE......Page 11
FOR STUDENTS: HOW TO USE THIS PROBLEMTEXT......Page 15
1 INTERVALS......Page 17
2 TOPOLOGY OF THE REAL LINE......Page 21
3. CONTINUOUS FUNCTIONS FROM R TO R......Page 25
4. SEQUENCES OF REAL NUMBERS......Page 33
5. CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM......Page 43
6. COMPACTNESS AND THE EXTREME VALUE THEOREM......Page 49
7. LIMITS OF REAL VALUED FUNCTIONS......Page 55
8. DIFFERENTIATION OF REAL VALUED FUNCTIONS......Page 59
9. METRIC SPACES......Page 67
10. INTERIORS, CLOSURES, AND BOUNDARIES......Page 73
11. THE TOPOLOGY OF METRIC SPACES......Page 77
12. SEQUENCES IN METRIC SPACES......Page 81
13. UNIFORM CONVERGENCE......Page 85
14. MORE ON CONTINUITY AND LIMITS......Page 89
15. COMPACT METRIC SPACES......Page 99
16. SEQUENTIAL CHARACTERIZATION OF COMPACTNESS......Page 103
17. CONNECTEDNESS......Page 109
18. COMPLETE SPACES......Page 113
19. APPLICATIONS OF A FIXED POINT THEOREM......Page 117
20. VECTOR SPACES......Page 123
21. LINEARITY......Page 131
22. NORMS......Page 145
23. CONTINUITY AND LINEARITY......Page 153
24. THE CAUCHY INTEGRAL......Page 161
25. DIFFERENTIAL CALCULUS......Page 173
26. PARTIAL DERIVATIVES AND ITERATED INTEGRALS......Page 185
27. COMPUTATIONS IN Rn......Page 197
28. INFINITE SERIES......Page 211
29. THE IMPLICIT FUNCTION THEOREM......Page 225
APPENDIX A QUANTIFIERS......Page 235
APPENDIX B SETS......Page 237
APPENDIX C SPECIAL SUBSETS OF R......Page 241
APPENDIX D LOGICAL CONNECTIVES......Page 243
APPENDIX E WRITING MATHEMATICS......Page 249
APPENDIX F SET OPERATIONS......Page 253
APPENDIX G ARITHMETIC......Page 259
APPENDIX H ORDER PROPERTIES OF R......Page 263
APPENDIX I. NATURAL NUMBERS AND MATHEMATICAL INDUCTION......Page 265
APPENDIX J. LEAST UPPER BOUNDS AND GREATEST LOWER BOUNDS......Page 269
APPENDIX K. PRODUCTS, RELATIONS, AND FUNCTIONS......Page 275
APPENDIX L. PROPERTIES OF FUNCTIONS......Page 279
APPENDIX M. FUNCTIONS WHICH HAVE INVERSES......Page 283
APPENDIX N. PRODUCTS......Page 287
APPENDIX O. FINITE AND INFINITE SETS......Page 289
APPENDIX P. COUNTABLE AND UNCOUNTABLE SETS......Page 293
APPENDIX Q. SOLUTIONS TO EXERCISES......Page 297
Bibliography......Page 371
Index......Page 373