hw11 - ± f x n-f y n)± ≥ ε Hint First prove ²⇒...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 11 Due date: Nov 12 (Wed) Exercises from the textbook. 5.4: 6; 7; 8; 15 6.1: 2; 7; 8(b)(d); 12 Regarding questions that have hints in the back of the textbook : Your score will be 0, if you just copy the hint. Out-of-textbook exercises. 1. Let D R and f D R . Prove the following sequential criteria for uniform continuity and its negation. (a) f is uniformly continuous on D ⇐⇒ for any sequences ( x n ) , ( y n ) D with x n - y n 0, we have f ( x n ) - f ( y n ) 0. (b) f is not uniformly continuous on D ⇐⇒ there are sequences ( x n ) , ( y n ) D with x n - y n 0 and ε > 0 such that
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Unformatted text preview: ± f ( x n ) -f ( y n )± ≥ ε . Hint : First prove ²⇒ direction for (a) and (b). This automatically implies ⇐² direction for both parts (explain why). 2. Let D = ( a,c ] ∪ [ b,d ) such that a < b and c < d , where d could be ∞ . Let f ∶ D → R and show that if f is uniformly continuous on ( a,c ] and f is uniformly continuous on [ b,d ) , then f is uniformly continuous on the whole D . caution : You have to consider the following cases separately: c < b , c = b and b < c ....
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