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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.
- Fall '08