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Unformatted text preview: ( x n ) has a cluster point in K x . (c) Find a closed set N in R 2 such that N x is not closed (in R ) . 3. Suppose K 1 ;K 2 ;:::;K m are compact sets. prove that K = K 1 [ K 2 [ ::: [ K m is compact in two ways: (a) Start with a cover for K and &nd a &nite subcover. (b) Start with a sequence in K and &nd a cluster point. 4. Problems on Continuity: (a) Let f : [0 ; 1] ! R be de&ned as f ( x ) = ² x if x 2 Q if x 62 Q : Use the & ² ± de&nition to prove that f is continuous at x = 0 : Is it continuous anywhere else? (b) Let f : [0 ; 1] ! R be de&ned as f ( x ) = ² 1 if x 2 Q if x 62 Q : Prove that f is not continuous anywhere. 5. 1...
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This note was uploaded on 06/25/2010 for the course MATH MATH 301 taught by Professor Alberterkip during the Fall '08 term at Sabancı University.
 Fall '08
 ALBERTERKIP
 Math, Integers

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