mathAnalysis6-07

mathAnalysis6-07 - ARE211, Fall 2007 ANALYSIS6: TUE, SEP...

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ARE211, Fall 2007 ANALYSIS6: TUE, SEP 18, 2007 PRINTED: OCTOBER 17, 2007 (LEC# 6) Contents 1. Analysis (cont) 1 1.9. Continuous Functions 2 1.9.1. Continuous functions on compact sets attain extreme values 4 1.10. Upper and Lower Hemi continuous correpondences 9 1.10.1. Alternative defns of upper hemicontinuity 11 1.10.2. Alternative de±nitions of lower hemicontinuity 13 1. Analysis (cont) Before proceeding to the next topic, we’ll prove a completely obvious Fact about sequences and subsequences. (We could call it a Lemma, but that’s glorifying it.) Recall that to show that ( t n ) is a subsequence of ( x n ), you always have to show the existence of a strictly increasing function τ : N N such that for all n N , t n = x τ ( n ) . Now note: Fact: : If τ : N N is a strictly increasing mapping, then for all n , τ ( n ) n . To prove this, we’ll argue by induction. 1

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2 ANALYSIS6: TUE, SEP 18, 2007 PRINTED: OCTOBER 17, 2007 (LEC# 6) Initial step: τ (1) 1 (duh). Inductive step: suppose that τ ( n ) n ; then τ ( n + 1) n + 1. Proof of the Inductive step: Let τ ( n ) = k N , k > = n . since τ ( n +1) > τ ( n ) and τ ( n +1) N , τ ( n + 1) k + 1 n + 1. 1.9. Continuous Functions A function is continuous if it maps nearby points to nearby points. Draw the graph without taking pen oF paper. Graph is connected. ±ormally: Defnition: Consider f : X R k . ±ix x 0 X . The function f is continuous at x 0 if whenever { x m } m =1 is a sequence in X which converges to x 0 , then { f ( x m ) } m =1 converges to f ( x 0 ). The function f is continuous if it is continuous at x , for every x X . While this de²nition seems obvious and intuitive, all is not what it seems to be. We’ve noticed ad nauseum that whether or not a sequence converges depends on the metric and the universe. In this section, we now have two diFerent sequences, one in the domain and one in the range; so we have to worry about what kinds of things converge in the domain and what kinds of things converge in the range. So we have roughly double the number of counter-intuitive possibilities. To see what kinds of things can happen, consider the following question: Question: let f : X R k and let X be endowed with the discrete metric. What can we say about the continuity of f ? Answer: f is continuous.
ARE211, Fall 2007 3 In other words, if X is endowed with the discrete metric, then every function is continuous! The reason for this is that the discrete metric makes convergence an extremely stringent requirement: ( x n ) converges to x , iF, eventually, the sequence is constant at x . But whenever this stringent condition is satis±ed, the resulting sequence in the range, { f ( x n ) } is eventually constant at f ( x ).

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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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mathAnalysis6-07 - ARE211, Fall 2007 ANALYSIS6: TUE, SEP...

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