ECEN601_hw4_pfister - f-1 is not continuous. [You only need...

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ECEN 601: Assignment 4 Problems: 1. Prove that if a Cauchy sequence in a metric space (not assumed to be complete) has a limit point, then it has a limit. [A point x is a limit point of { x n } if every neighborhood of x contains inFnitely many elements of the sequence.] 2. Call two metrics d 1 and d 2 on the same set M equivalent if there exist positive constants c 1 ,c 2 such that d 1 ( x,y ) c 2 d 2 ( x,y ) and d 2 ( x,y ) c 1 d 1 ( x,y ) for all x and y in M . Prove that x n x in d 1 -metric if and only if x n x in d 2 -metric, if d 1 and d 2 are equivalent. 3. If f : M N and g : N P are continuous, prove g f : M P is continuous. Given an example where g f is continuous but g and f are not. 4. Let a be a Fxed real number with 1 < a < 3. Prove that the mapping f ( x ) = ( x/ 2) + ( a/ 2 x ) satisFes the hypotheses of the contractive mapping principle on the domain (1 , ). What is the Fxed point? 5. Show that f : [0 , 2 π ) S deFned by f ( t ) = (cos t, sin t ), where S is the unit circle in R 2 , is one-to-one, onto, and continuous, but
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Unformatted text preview: f-1 is not continuous. [You only need to argue that f is one-to-one and onto. However, you need to actually show the last two facts.] 6. If f satisFes the hypotheses of the contractive mapping principle and x 1 is any point in M , show that d ( x 1 ,x ) d ( x 1 ,f ( x 1 )) / (1 r ) where x is the Fxed point. Informally, this says that if f ( x 1 ) is close to x 1 , then x 1 is close to the Fxed point (but r must not be too close to 1 for this to be a good estimate). Optional Problems: 1. If f : M R n and g : M R are continuous, prove g f : M R n is continuous. [or m M , the function g f is equal to g ( m ) f ( m ).] 2. Give an example of a continuous function f : M N that does not take Cauchy sequences in M to Cauchy sequences in N . 1...
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This note was uploaded on 09/26/2011 for the course ECEN 601 taught by Professor Staff during the Fall '08 term at Texas A&M.

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