104hw7sum06 - MATH 104, SUMMER 2006, HOMEWORK 7 SOLUTION...

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Unformatted text preview: MATH 104, SUMMER 2006, HOMEWORK 7 SOLUTION BENJAMIN JOHNSON Due July 26 Assignment: Section 19: 19.2(b), 19.6, 19.9 Section 20: 20.16, 20.18 Section 19 19.2 Prove that each of the following functions are uniformly continuous on the indicated set by directly verifying the- property. (b) f ( x ) = x 2 on [0 , 3] Proof. Let a [0 , 3]. Let R . Assume > 0. Choose = min { 1 , 7 } . Then > 0. Let x , y [0 , 3]. Assume | x- y | < . Since | x- y | < 1, we have x- 1 y x + 1, and hence 0 x + y x + ( x + 1) 3 + 3 + 1 = 7. I.e. | x + y | < 7. Also | x- y | < 7 . So... | f ( x )- f ( y ) | = | x 2- y 2 | = | x + y | | x- y | < 7 7 = Thus f is uniformly continuous on [0 , 3]. 19.6 (a) Let f ( x ) = x for x 0. Show that f is unbounded on (0 , 1] but that f is nevertheless uniformly continuous on (0 , 1]. Compare with Theorem 19.6. Proof. f ( x ) = 1 2 x , which is unbounded on (0 , 1]. [For proof... Let M R . Choose x = 1 4( | M | + 1)) 2 . Then x (0 , 1], and f ( x ) = 1 2 q 1 4( | M | + 1) 2 = | M | + 1 > M . So ( M R )( x (0 , 1])( f ( x ) > M ), and f is unbounded above in (0 , 1].] Nevertheless,...
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104hw7sum06 - MATH 104, SUMMER 2006, HOMEWORK 7 SOLUTION...

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