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If $ f \in C_0^\infty=\{ g: g\in C^\infty, \lim_{|x|\rightarrow \infty}g(x)=0\}$, then is $f$ uniformly continuous on $\mathbb R$? ($ f : \mathbb R \to \mathbb R $)

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    This is true. Hint: You can split $f$ up into a part where you can control it and a different part where it is very small2012-06-10
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    Doesn't this just follow from the fact that $f$ has compact support, is continuous on that compact set, and hence uniformly continuous on that set (and hence all of $\mathbb{R}$, since $f\equiv0$ outside of the set.)2012-06-10
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    Doesn't the $0$ in the subscript mean compact support? Then it follows from being continuous on a compact set...2012-06-10
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    possible duplicate of [Is an increasing, bounded and continuous function on $\[a,+\infty)$ uniformly continuous?](http://math.stackexchange.com/questions/105310/is-an-increasing-bounded-and-continuous-function-on-a-infty-uniformly-con)2012-06-10
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    Note, the same could be said about $f\in C^{1}_{0}$.2012-06-10
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    The question itself is not an exact duplicate but all you need is that $\lim_{x \rightarrow \infty}f(x)$ exists. Which reduces this to the other question.2012-06-10
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    Thank you. I think it also holds for just $f \in C_0^1$.2012-06-10
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    @Patch I think the subscript $0$ denotes that $\lim_{|x|\to \infty} f(x) =0$, rather than compact support.2012-06-10
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    Oh. I'd never seen that notation before.2012-06-10
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    @Misaj It is prefered that when using non-standard notation, to give an definition or explainations in words what it means. Notice that sometimes different people using different notations, so not each notation is generally understood by all.2012-06-10
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    People from PDE use $C_0^\infty$ to denote compactly supported functions. People from harmonic analysis (like Rudin) use $C_c$ to denote continuous functions with compact support, while the subscript 0 is used for "vanishing at infinity".2012-06-10
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    I do not think that this is a duplicate of that question.2012-06-10

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HINTs

  1. A continuous function on a compact interval is uniformly continuous.
  2. $\lim_{|x| \to \infty} f(x) = 0$ means that $\forall \epsilon...$
  3. Split up the domain to use these two properties.