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Unformatted text preview: ) that converges to x , we have lim f ( x n ) = f ( x ) , then f is said to be continuous at x dom( f ) . (b) (10 points) Prove that the function f below is continuous at x = 0 . f ( x ) = x cos( 1 x 2 ) , if x 6 = 0 , if x = 0 . Proof. Suppose ( x n ) is any sequence in dom( f ) = R that converges to x = 0 . Then f ( x n ) = x n cos( 1 x 2 n ) , if x n 6 = 0 , if x n = 0 . In either case, x n  f ( x n )  x n  , for all n , since  cos( 1 x 2 )  1 for all x . By squeeze theorem, lim n f ( x n ) = 0 = f (0) , since lim  x n  = lim( x n  ) = 0 . Therefore, f is continuous at x = 0 . 2...
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 Spring '10
 GuoliangWu
 Math

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