## Problems in Real Analysis: Advanced Calculus on the Real AxisProblems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis. Key features: *Uses competition-inspired problems as a platform for training typical inventive skills; *Develops basic valuable techniques for solving problems in mathematical analysis on the real axis and provides solid preparation for deeper study of real analysis; *Includes numerous examples and interesting, valuable historical accounts of ideas and methods in analysis; *Offers a systematic path to organizing a natural transition that bridges elementary problem-solving activity to independent exploration of new results and properties. |

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### Contents

Sequences | 3 |

12 Introductory Problems | 7 |

13 Recurrent Sequences | 18 |

14 Qualitative Results | 30 |

15 Hardys and Carlemans Inequalities | 45 |

16 Independent Study Problems | 51 |

Series | 58 |

22 Elementary Problems | 66 |

65 Independent Study Problems | 285 |

Inequalities and Extremum Problems | 288 |

72 Elementary Examples | 290 |

73 Jensen Young Holder Minkowski and Beyond | 294 |

74 Optimization Problems | 300 |

75 Qualitative Results | 305 |

76 Independent Study Problems | 308 |

Antiderivatives Riemann Integrability and Applications | 311 |

23 Convergent and Divergent Series | 73 |

24 Infinite Products | 86 |

25 Qualitative Results | 89 |

26 Independent Study Problems | 110 |

Limits of Functions | 115 |

32 Computing Limits | 118 |

33 Qualitative Results | 124 |

34 Independent Study Problems | 133 |

Qualitative Properties of Continuous and Differentiable Functions | 136 |

Continuity | 139 |

42 Elementary Problems | 144 |

43 The Intermediate Value Property | 147 |

44 Types of Discontinuities | 151 |

45 Fixed Points | 154 |

46 Functional Equations and Inequalities | 163 |

47 Qualitative Properties of Continuous Functions | 169 |

48 Independent Study Problems | 177 |

Differentiability | 182 |

52 Introductory Problems | 198 |

53 The Main Theorems | 218 |

54 The Maximum Principle | 235 |

55 Differential Equations and Inequalities | 238 |

56 Independent Study Problems | 252 |

Applications to Convex Functions and Optimization | 260 |

Convex Functions | 263 |

62 Basic Properties of Convex Functions and Applications | 265 |

63 Convexity versus Continuity and Differentiability | 273 |

64 Qualitative Results | 278 |

Antiderivatives | 313 |

82 Elementary Examples | 315 |

83 Existence or Nonexistence of Antiderivatives | 317 |

84 Qualitative Results | 319 |

85 Independent Study Problems | 324 |

Riemann Integrability | 325 |

92 Elementary Examples | 329 |

93 Classes of Riemann Integrable Functions | 337 |

94 Basic Rules for Computing Integrals | 339 |

95 Riemann Iintegrals and Limits | 341 |

96 Qualitative Results | 351 |

97 Independent Study Problems | 367 |

Applications of the Integral Calculus | 373 |

102 Integral Inequalities | 374 |

103 Improper Integrals | 390 |

104 Integrals and Series | 402 |

105 Applications to Geometry | 406 |

106 Independent Study Problems | 409 |

Appendix | 415 |

Basic Elements of Set Theory | 417 |

A2 Finite Countable and Uncountable Sets | 418 |

Topology of the Real Line | 419 |

B2 Some Distinguished Points | 420 |

Glossary | 421 |

437 | |

443 | |