Author	Abbott, Stephen. author
Title	Understanding Analysis [electronic resource] / by Stephen Abbott
Imprint	New York, NY : Springer New York : Imprint: Springer, 2001
Connect to	http://dx.doi.org/10.1007/978-0-387-21506-8
Descript	XII, 260 p. online resource

Understanding Analysis outlines an elementary, one-semester course designed to expose students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on the questions that give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Are derivatives continuous? Are derivatives integrable? Is an infinitely differentiable function necessarily the limit of its Taylor series? In giving these topics center stage, the hard work of a rigorous study is justified by the fact that they are inaccessible without it

1 The Real Numbers -- 1.1 Discussion: The Irrationality of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaSbdaGcaa % qaaiaaikdaaSqabaaaaa!3794! $$\sqrt 2 $$ -- 1.2 Some Preliminaries -- 1.3 The Axiom of Completeness -- 1.4 Consequences of Completeness -- 1.5 Cantorโ{128}{153}s Theorem -- 1.6 Epilogue -- 2 Sequences and Series -- 2.1 Discussion: Rearrangements of Infinite Series -- 2.2 The Limit of a Sequence -- 2.3 The Algebraic and Order Limit Theorems -- 2.4 The Monotone Convergence Theorem and a First Look at Infinite Series -- 2.5 Subsequences and the Bolzano-Weierstrass Theorem -- 2.6 The Cauchy Criterion -- 2.7 Properties of Infinite Series -- 2.8 Double Summations and Products of Infinite Series -- 2.9 Epilogue -- 3 Basic Topology of R -- 3.1 Discussion: The Cantor Set -- 3.2 Open and Closed Sets -- 3.3 Compact Sets -- 3.4 Perfect Sets and Connected Sets -- 3.5 Baireโ{128}{153}s Theorem -- 3.6 Epilogue -- 4 Functional Limits and Continuity -- 4.1 Discussion: Examples of Dirichlet and Thomae -- 4.2 Functional Limits -- 4.3 Combinations of Continuous Functions -- 4.4 Continuous Functions on Compact Sets -- 4.5 The Intermediate Value Theorem -- 4.6 Sets of Discontinuity -- 4.7 Epilogue -- 5 The Derivative -- 5.1 Discussion: Are Derivatives Continuous? -- 5.2 Derivatives and the Intermediate Value Property -- 5.3 The Mean Value Theorem -- 5.4 A Continuous Nowhere-Differentiable Function -- 5.5 Epilogue -- 6 Sequences and Series of Functions -- 6.1 Discussion: Branching Processes -- 6.2 Uniform Convergence of a Sequence of Functions -- 6.3 Uniform Convergence and Differentiation -- 6.4 Series of Functions -- 6.5 Power Series -- 6.6 Taylor Series -- 6.7 Epilogue -- 7 The Riemann Integral -- 7.1 Discussion: How Should Integration be Defined? -- 7.2 The Definition of the Riemann Integral -- 7.3 Integrating Functions with Discontinuities -- 7.4 Properties of the Integral -- 7.5 The Fundamental Theorem of Calculus -- 7.6 Lebesgueโ{128}{153}s Criterion for Riemann Integrability -- 7.7 Epilogue -- 8 Additional Topics -- 8.1 The Generalized Riemann Integral -- 8.2 Metric Spaces and the Baire Category Theorem -- 8.3 Fourier Series -- 8.4 A Construction of R From Q