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 |
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 Cantorโs 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 Baireโs 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 Lebesgueโs 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