Understanding Analysis

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Introduction to the Problems in Analysis outlines an elementary, one semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic 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 questions which 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? Can the rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it. Subjects of the book : Mathematics Contents of the book : Preface v 1 The Real Numbers 1 1.1 Discussion: The Irrationality of v2 1.2 Some Preliminaries 1.3 The Axiom of Completeness 1.4 Consequences of Completeness 1.5 Cantor’s Theorem 1.6 Epilogue 2 Sequences and Series 35 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 75 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 Baire’s Theorem 3.6 Epilogue 4 Functional Limits and Continuity 99 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 129 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 151 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 183 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 Lebesgue’s Criterion for Riemann Integrability 7.7 Epilogue 8 Additional Topics 213 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 Bibliography Index