hw2 - c ∈[0 1 such that f c = c This c is called a fixed...

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Math 410 - Homework 2 - Due Tuesday June 14 (1) Let s n = n k =1 1 /k . Use the principle of Mathematical Induction to prove that for a natural numbers n , s 2 n 1 + n/ 2. (2) Give an example of nested and open intervals I n = ( a n ,b n ) such that their intersection n =1 I N is empty. (3) Prove (using ± - N ) that f ( x ) = x is continuous on [0 , ). (4) Prove (using ± - N ) that f ( x ) = | x | is continuous on all of R . (5) Assume that f : [0 , 1] [0 , 1] is continuous and show that there is a
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Unformatted text preview: c ∈ [0 , 1] such that f ( c ) = c . This c is called a fixed point for f . [Hint: consider g ( x ) = f ( x )-x and investigate the endpoints.] (6) Is f ( x ) = x 2 uniformly continuous on R ? Is it uniformly continuous on (0 , 1)? Show why or why not. (7) Use the ±-δ definition to show that f ( x ) = x 3 is continuous at all points in R ....
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This note was uploaded on 02/09/2012 for the course MATH 410 taught by Professor Staff during the Summer '08 term at Maryland.

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