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Math Help - Uniform Continuity

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    Uniform Continuity

    Let f(x) be continous on the real number line. Suppose that lim(x-> infinity) f(x) = 0 and lim(x-> negative infinity) f(x) = 0. Show that f(x) is unformly continous or else give a counterexample.
    Last edited by seams192; May 13th 2010 at 11:03 AM. Reason: Shortening the problem
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    Quote Originally Posted by seams192 View Post
    Let f(x) be continuous on the whole real number line. Suppose that lim(x-> infinity) f(x) = 0 and lim(x-> negative infinity) f(x) = 0. Either show that f(x) is uniformly continuous or else give a counterexample.

    ----------------------------
    A perfect example of this would be a function like 1/(x^2+1) which is uniformly continuous. I would have difficulty proving it in the general case, but am also blind to any counterexamples that could exist.

    Any thoughts?
    More generally, if \lim_{x\to\infty}f(x)=L_1,\lim_{x\to-\infty}f(x)=L_2<\infty and f is continuous then it's uniformly continuous.

    Start like this, for some \varepsilon>0 choose A>0 such that |f(x)-L_1|<\frac{\varepsilon}{2},\text{ }x\in[A,\infty) AND |f(x)-L_2|<\frac{\varepsilon}{2},\text{ }x\in(-\infty,-A].

    You clearly have then that |f(x_1)-f(x_2)|<\varepsilon,x_1,x_2\in\mathbb{R}-[-A,A] and you surely have that f is unif. cont. on [-A,A]. See if you piece the riece together.
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    Why does it follow that it is uniformly continuous on the interval [-M,M]? I can see the intervals (-inf,-M] and [M,inf) , since the delta neighborhood of x values need not depend on x, only epsilon as the slope varys less and less towards the ends of the domain.
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    Quote Originally Posted by seams192 View Post
    Why does it follow that it is uniformly continuous on the interval [-M,M]?
    Any function that is continuous on a compact set of real numbers is uniformly continuous. [-M,M] is compact.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by seams192 View Post
    Why does it follow that it is uniformly continuous on the interval [-M,M]? I can see the intervals (-inf,-M] and [M,inf) , since the delta neighborhood of x values need not depend on x, only epsilon as the slope varys less and less towards the ends of the domain.
    The specific case we need of the theorem says that

    If f:E\to\mathbb{R} is continuous with E\subseteq\mathbb{R} compact then f is in fact uniformly continuous
    Are you aware of this theorem?
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    Ohh right. The Uniform Continuity theorem. Thanks!
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