# hw3 - tinuous In other words if f g are uniformly...

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Math 125A, Fall 2009 Homework 3 Due Date: October 16, 2009 Problem 1: Let f be a uniformly continuous function. Prove that if ( x n ) , ( y n ) are sequences such that | x n - y n | → 0, then | f ( x n ) - f ( y n ) | → 0 also. Problem 2: Determine if the function is uniformly continuous on the speciﬁed set and justify your answer. (a) f ( x ) = sin( e x 2 + cos(3 x 2 + 5)) on [0 , 10]. (b) f ( x ) = ln( x ) on (0 , 1). (c) f ( x ) = x ln( x ) on (0 , 1). Problem 3: Determine whether or not the statement is true. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If f is bounded and continuous, then f is uniformly continuous. (b) If f is uniformly continuous on its domain, then f is bounded. (c) If f is uniformly continuous on a bounded set, then f is bounded. Problem 4: Prove that the composition of uniformly continuous functions is uniformly con-
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Unformatted text preview: tinuous. In other words, if f, g are uniformly continuous functions, then prove that h = g ◦ f is also uniformly continuous. Problem 5: (a) Let f : R → R be continuous, and suppose also that it is periodic , i.e., that there is a positive number c such that f ( x + c ) = f ( x ) for all x . Prove that f is uniformly continuous. (b) Thus we conclude that the function sin x is uniformly continuous. Prove, however, that sin( x 2 ) is not uniformly continuous on R . Problem 6: Let f be a continuous function for which lim x →∞ f ( x ) and lim x →-∞ f ( x ) both exist and are ﬁnite. Prove that f is bounded....
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