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Unformatted text preview: Solutions to Homework 1 January 16, 2009 17.3: Assume that log e , cos , sin , and x p are all continuous functions. a: Continuity through Theorem 17.4(i) and 17.5 b: Continuity through Theorem 17.5 on cos 6 and sin 2 , then 17.4(i) and finaly 17.5 again 17.4: Prove that the function ( x ) 1 / 2 is continuous on [0 , ) Proof: case 1: x = 0: for any > , < 2 s.t. for all  x  < ,  ( x ) 1 / 2 (0) 1 / 2  = ( x ) 1 / 2 < ( ) 1 / 2 = case 2: x (0 , ): for any > ( x ) 1 / 2 such that  x x  < ,  ( x ) 1 / 2 ( x ) 1 / 2  =  ( x ) 1 / 2 ( x ) 1 / 2  1  ( x ) 1 / 2 + ( x ) 1 / 2   ( x ) 1 / 2 + ( x ) 1 / 2  =  x x  ( x ) 1 / 2 + ( x ) 1 / 2 x x ( x ) 1 / 2 < ( x ) 1 / 2 ( x ) 1 / 2 ( x ) 1 / 2 = 17.9a Prove that f ( x ) = x 2 is continuous at x = 2 using property: Note that if  x 2  < 1 then by the triangle inequality  x  < 1 + 2 = 3 and using the triangle eniquality again we get  x + 2   x...
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 Spring '09
 Continuity

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