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# hw19solutions - Math 521 Advanced Calculus I Instructor J...

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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 ε > 0, and we need to show that there is an x ∈ E so that | f ( x )- f ( c ) | < ε . That is, we are showing that there is a member of f ( E ) in V ε ( f ( c )). By continuity, there exists a δ > 0 so that if | x...
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