hw19solutions - Math 521 - Advanced Calculus I Instructor:...

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Unformatted text preview: Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: March 24, 2010 Assignment 19 1. If f : R R is continuous, prove that f ( E ) f ( E ) for every set E R . ( E denotes the closure of E as defined on exam 1). Show, by an example, that f ( E ) can be a proper subset of f ( E ). For the latter portion, we know that we must choose E to be non-compact. One example is f ( x ) = 1 / (1 + x 2 ). Then f ((1 , )) = (0 , 1 / 2). But (1 , ) = [1 , ), but f ([1 , )) = (0 , 1 / 2] ( [0 , 1 / 2] = (0 , 1 / 2] . That f ( E ) f ( E ) is trivial. Thus, to show the inclusion, we let c be a limit point of E with c 6 E and need to show that f ( c ) f ( E ). To do this, we shall need to show that f ( c ) is a limit point of f ( E ). Fix > 0, and we need to show that there is an x E so that | f ( x )- f ( c ) | < . That is, we are showing that there is a member of f ( E ) in V ( f ( c )). By continuity, there exists a > 0 so that if | x...
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This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.

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