Introduction to Mathematical Analysis

ISBN-13: 9780710075291 / Angielski / Miękka / 1973 / 83 str.

C. R. J. Clapham

Opis

I have tried to provide an introduction, at an elementary level, to some of the important topics in real analysis, without avoiding reference to the central role which the completeness of the real numbers plays throughout. Many elementary textbooks are written on the assumption that an appeal to the complete- ness axiom is beyond their scope; my aim here has been to give an account of the development from axiomatic beginnings, without gaps, while keeping the treatment reasonably simple. Little previous knowledge is assumed, though it is likely that any reader will have had some experience of calculus. I hope that the book will give the non-specialist, who may have considerable facility in techniques, an appreciation of the foundations and rigorous framework of the mathematics that he uses in its applications; while, for the intending mathe- matician, it will be more of a beginner's book in preparation for more advanced study of analysis. I should finally like to record my thanks to Professor Ledermann for the suggestions and comments that he made after reading the first draft of the text.

Spis treści

Content.- 1. Axioms for the Real Numbers.- 1 Introduction.- 2 Fields.- 3 Order.- 4 Completeness.- 5 Upper bound.- 6 The Archimedean property.- Exercises.- 2. Sequences.- 7 Limit of a sequence.- 8 Sequences without limits.- 9 Monotone sequences.- Exercises.- 3. Series.- 10 Infinite series.- 11 Convergence.- 12 Tests.- 13 Absolute convergence.- 14 Power series.- Exercises.- 4. Continuous Functions.- 15 Limit of a function.- 16 Continuity.- 17 The intermediate value property.- 18 Bounds of a continuous function.- Exercises.- 5. Differentiable Functions.- 19 Derivatives.- 20 Rolle’s theorem.- 21 The mean value theorem.- Exercises.- 6. The Riemann Integral.- 22 Introduction.- 23 Upper and lower sums.- 24 Riemann-integrable functions.- 25 Examples.- 26 A necessary and sufficient condition.- 27 Monotone functions.- 28 Uniform continuity.- 29 Integrability of continuous functions.- 30 Properties of the Riemann integral.- 31 The mean value theorem.- 32 Integration and differentiation.- Exercises.- Answers to the Exercises.