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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- Fall '07
- Calculus, Continuous function, Uniform continuity, uniformly continuous function, uniformly continuous functions