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Unformatted text preview: 3: Let f : [ a,b ] R be continuous. Show the followings: a) For any > 0, there is a > 0 such that if x, x [ a,b ] with  x x  < , then  f ( x )f ( x )  < . b) Give an example of continuous f : (0 , 1) R for which the statement in part (a) fails. Justify your answer. 4: Let f : (0 , 1) R be increasing and bounded. Show that lim x f ( x ) exists. Important results: Archimedean property, monotone and bounded sequences are convergent, closed and bounded sets are compact, Extreme Value Theorem, Intermediate Value Theorem, continuous maps on compacta are uniformly continuous. 1...
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This note was uploaded on 11/18/2011 for the course MAP 4426 taught by Professor Tamasan during the Fall '10 term at University of Central Florida.
 Fall '10
 Tamasan

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