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Unformatted text preview: Analysis Course C9 T. W. K¨orner December 19, 1998 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This document is written in L A T E X2e and stored in the file labelled ~twk/1B/Anal.tex on emu in (I hope) read permitted form. My email address is twk@dpmms . Contents 1 Why do we bother? 2 2 The axiom of Archimedes 3 3 The Bolzano–Weierstrass theorem 4 4 Higher dimensions 8 5 Open and closed sets 10 6 The central theorems of analysis 13 7 Differentiation 16 8 Local Taylor theorems 20 9 Riemann integration 24 10 Further remarks on integration 28 11 Metric spaces 31 1 12 A look to the future 34 13 Completeness 37 14 The uniform metric 40 15 The contraction mapping theorem 44 16 Green’s function 47 17 Further reading 50 18 First Sheet of Exercises 52 19 Second Sheet of Exercises 58 20 Third Sheet of Exercises 62 21 Fourth Sheet of Exercises 66 1 Why do we bother? It is surprising how many people think that that analysis consists in the difficult proofs of obvious theorems. All we need know, they say, is what a limit is, the definition of continuity and the definition of the derivative. All the rest is ‘intuitively clear’. If pressed they will agree that these definitions apply as much to the rationals Q as to the real numbers R . They then have to explain the following interesting example. Example 1.1. If f : Q → Q is given by f ( x ) = 1 if x 2 < 2 , f ( x ) = 1 otherwise, then (i) f is continuous function with f (0) = 1 , f (2) = 1 yet there does not exist a c with f ( c ) = 0 , (ii) f is a differentiable function with f ( x ) = 0 for all x yet f is not constant. What is the difference between R and Q which makes calculus work on one even though it fails on the other. Both are ‘ordered fields’, that is, both 2 support operations of ‘addition’ and ‘multiplication’ together with a relation ‘greater than’ (‘order’) with the properties that we expect. If the reader is interested she will find a complete list of the appropriate axioms in texts like the altogether excellent book of Spivak [8] and its many rather less excellent competitors, but, interesting as such things may be, they are irrelevant to our purpose which is not to consider the shared properties of R and Q but to identify a difference between the two systems which will enable us to exclude the possibility of a function like that of Example 1.1 for functions from R to R . To state the difference we need only recall a definition from the beginning of the course C5/6....
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 Fall '11
 DrewArmstrong
 The Land

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