Stephen Abbott (1) (1964–)

skimmed, imo, good for getting an idea of what analysis is and its general "flavor", however for me it was not rigorous for self studying because descriptions focused on being intuitive rather than precise. I will be using the tao book along with baby rudin instead :|

If you have ever wanted to push the envelope for your mathematical learning, I suggest reading this book. It is an overview of mathematical analysis, involving rigorous proofs. The book is small, but it is filled to the brim with knowledge. A must for any serious student of mathematics.

Indeholder "Preface", " The Main Objectives", " The Structure of the Book", " Building a Course", " The Audience", " Acknowledgments", "1 The Real Numbers", " 1.1 Discussion: The Irrationality of sqrt(2)", " 1.2 Some Preliminaries", " 1.3 The Axiom of Completeness", " 1.4 Consequences of Completeness", " 1.5 Cardinality", " 1.6 Cantor's Theorem", " 1.7 Epilogue", "2 Sequences and Series", " 2.1 Discussion: Rearrangements of Infinite Series", " 2.2 The Limit of a Sequence", " 2.3 The show more 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 Baire's Theorem", " 3.6 Epilogue", "4 Functional Limits and Continuity", " 4.1 Discussion: Examples of Dirichlet and Thomae", " 4.2 Functional Limits", " 4.3 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 Theorems", " 5.4 A Continuous Nowhere-Differentiable Function", " 5.5 Epilogue", "6 Sequences and Series of Functions", " 6.1 Discussion: The Power of Power Series", " 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 The Weierstrass Approximation Theorem", " 6.8 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 Lebesgue'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 Euler's Sum", " 8.4 Inventing the Factorial Function", " 8.5 Fourier Series", " 8.6 A Construction of R From Q", "Bibliography", "Index".

Standardlærebog i matematik første år på universitetet. Her er nogle af de funktioner, der gav grå hår i hovedet, da man formaliserede funktionsbegrebet, fx en som Carl Johannes Thomae opdagede i 1875. f(x) er 1, hvis x er 0, 0 hvis x er irrationel og 1/n, hvis x er i Q (fraregnet 0) og er på uforkortelig form m/n. Det lyder jo fredeligt nok, men det betyder at f er kontinuert i x, hvis x er irrationel og diskontinuert hvis x er rationel. Den har mange navne, fx the popcorn function og er periodisk med periode 1. show less