mathAnalysis6-07-draft

# mathAnalysis6-07-draft - P r e l i m i n a r y d r a f t o...

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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #6: TUE, SEP 18, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS6) 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 8 1.10.1. Alternative defns of upper hemicontinuity 10 1.10.2. Alternative definitions of lower hemicontinuity 11 1. Analysis (cont) Before proceeding to the next topic, well prove a completely obvious Fact about sequences and subsequences. (We could call it a Lemma, but thats glorifying it.) Recall that to show that ( y n ) is a subsequence of ( x n ), you always have to show the existence of a strictly increasing function : N N such that for all n N , y n = x ( n ) . Now note: Fact: : If : N N is a strictly increasing mapping, then for all n , ( n ) n . To prove this, well argue by induction. Initial step: (1) 1 (duh). 1 2 LECTURE #6: TUE, SEP 18, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS6) Inductive step: suppose that ( n ) n ; then ( n + 1) n + 1. Proof of the Inductive step: Let ( n ) = k N , k > = n . since ( n +1) > ( n ) and ( n +1) N , ( 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 off paper. Graph is connected. Formally: Definition: Consider f : X R k . Fix x X . The function f is continuous at x if whenever { x m } m =1 is a sequence in X which converges to x , then { f ( x m ) } m =1 converges to f ( x ). The function f is continuous if it is continuous at x , for every x X . While this definition seems obvious and intuitive, all is not what it seems to be. Weve noticed ad nauseum that whether or not a sequence converges depends on the metric and the universe. In this section, we now have two different 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. 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 , iff, eventually, the sequence is constant at x . But whenever this stringent condition is satisfied, 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-draft - P r e l i m i n a r y d r a f t o...

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