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Unformatted text preview: Page 109 Problem 3. Let 0 < y < x . Then 1 1 + y 21 1 + x 2 = ( xy ) x + y (1 + y 2 )(1 + x 2 ) . Hence we need to show x + y (1 + y 2 )(1 + x 2 ) for proving f ( y )f ( x ) ( xy ). Now ( x1) 2 0 implies that x 1 2 + 1 2 x 2 . Similarly y 1 2 + 1 2 y 2 . Adding these two inequalities we get x + y 1+ 1 2 x 2 + 1 2 y 2 1+ x 2 + y 2 + x 2 y 2 = (1+ y 2 )(1+ x 2 ). Hence  f ( x )f ( y )   xy  . Now given > 0 we can take = to get from  xy  < that  f ( x )f ( y )  < , i.e., f is uniformly continuous. 1...
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This note was uploaded on 02/05/2012 for the course MATH 554 taught by Professor Girardi during the Fall '10 term at South Carolina.
 Fall '10
 Girardi

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