m409ss05_qk4

# m409ss05_qk4 - Math 409-300 (6/29/05) Name 1 Quiz 4...

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Math 409-300 (6/29/05) Name 1 Quiz 4 Instructions: Show all work in the space provided. Add separate sheets of paper if necessary. Due date: Wednesday, June 29. 1. Deﬁne the terms below. (a) (5 pts.) uniform continuity on a nonempty subset E R – p. 80. (b) (5 pts.) extension of a function – p. 82. 2. (15 pts.) Prove this: If f : [0 , ) R is continuous and if the lim x →∞ f ( x ) = L exists and is ﬁnite, then f is uniformly continuous on [0 , ). (Hint: look at your notes from 6/27/05.) Proof . Let ε > 0. Since f ( x ) L as x → ∞ , we can ﬁnd M > 0 such that when x > M we have | f ( x ) - L | < ε/ 2. With this M , we also note that f is continuous on the closed, bounded interval [0 , 2 M ], and is therefore uniformly continuous on [0 , 2 M ]. Thus for ε there is a δ 1 > 0 such that whenever two points x, t [0 , 2 M ] satisfy | x - t | < δ 1 , we have | f ( x ) - f ( t ) | < ε . Next, choose δ = min { M/ 2 , δ 1 } . Any two points x, t [0 , ) are either both in [0

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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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m409ss05_qk4 - Math 409-300 (6/29/05) Name 1 Quiz 4...

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