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Unformatted text preview: Math 28 Spring 2008: Exam 2 Instructions: Each problem is scored out of 10 points for a total of 40 points. You may not use any outside materials(eg. notes or books). You have 50 minutes to complete this exam. Problem 1. (a) Let f : A → R where A ⊂ R . State the definition for f to be uniformly continuous on A . (b) Which of the following functions are uniformly continuous on [0 , ∞ )? (i) f ( x ) = sin( x 2 ) (ii) f ( x ) = 1 x +1 Proof. (a) A function f : A → R is uniformly continuous on A if for every ² > 0 there exists a δ > 0 such that for every x,y ∈ A , | x- y | < δ implies that | f ( x )- f ( y ) | < ² . (b) (i) No. Let the sequence of points ( x n ) be defined by x n = 2 nπ and the sequence of points ( y n ) be defined by y n = 2 nπ . Then we have | f ( x n )- f ( y n ) | = fl fl fl fl f ( √ 2 nπ )- f ( r 2 nπ + π 2 ) fl fl fl fl = | sin(2 nπ )- sin(2 nπ + π/ 2) | = 1 . Also notice that | x n- y n | → 0 as n → ∞ . So for any ² ≤ 1 and for any δ > 0 there exists an N ∈ N such that | x n- y n | < δ , but we have that | f ( x N )- f ( y N ) |...
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This note was uploaded on 03/02/2010 for the course MATH 28 taught by Professor Staff during the Spring '08 term at UMass (Amherst).
- Spring '08