h9ra - Homework 9 Hctor Guillermo Cullar R e e os May 9,...

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Homework 9 ector Guillermo Cu´ ellar R´ ıos May 9, 2006 23.3 Determine which of the following continuous functions are uniformly con- tinuous on the given set. Justify your answers. (a) f ( x ) = e x x on [2 , 5] It is uniformly continuous since it is continuous on a compact set. (c) f ( x ) = x 2 + 3 x - 5 on [0 , 4] It is uniformly continuous since it is continuous on a compact set. (f) f ( x ) = 1 x 2 on (0 , ) It is not uniformly continuous since it can- not be extended to a function that is continuous on [0 , ] because lim x 0 f ( x ) = (g) f ( x ) = xsin ( 1 x ) on (0 , 1) It is uniformly continuous since it can be extended to a function that is contiuous on [0 , 1] by Example 21.5. 23.4 (a) f ( x ) = x 3 on [0 , 2] Proof. Given any ε > 0 we want to make | f ( x ) - f ( y ) | < ε . Now | f ( x ) - f ( y ) | = | x 3 - y 3 | = | x - y || x 2 + xy + y 2 | but | x 2 + xy + y 2 | ≤ 12, then we let δ = ε 12 and | x - y | < δ , we have | f ( x ) - f ( y ) | = | x - y
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This note was uploaded on 01/27/2010 for the course MATH-M 413 taught by Professor Michaeljolly during the Spring '08 term at Indiana.

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h9ra - Homework 9 Hctor Guillermo Cullar R e e os May 9,...

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