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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 ǫ > , ∃ δ < ǫ 2 s.t. for all  x −  < δ,  ( x ) 1 / 2 − (0) 1 / 2  = ( x ) 1 / 2 < ( δ ) 1 / 2 = ǫ case 2: x ∈ (0 , ∞ ): for any ǫ > ∃ δ ≤ ǫ ( x ) 1 / 2 such that ∀ x − x  < δ,  ( 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 < δ ( x ) 1 / 2 ≤ ǫ ( x ) 1 / 2 ( x ) 1 / 2 = ǫ 17.9a Prove that f ( x ) = x 2 is continuous at x = 2 using ǫ − δ 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, Continuous function, Irrational number, Sn

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