This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 counterintuitive 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 )....
View
Full
Document
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.
 Fall '07
 Simon

Click to edit the document details