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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1This is true. Hint: You can split $f$ up into a part where you can control it and a different part where it is very small – Listing Jun 10 '12 at 12:09
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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.) – Patch Jun 10 '12 at 12:12
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Doesn't the $0$ in the subscript mean compact support? Then it follows from being continuous on a compact set... – Seth Jun 10 '12 at 12:13
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possible duplicate of Is an increasing, bounded and continuous function on $[a,+\infty)$ uniformly continuous? – Listing Jun 10 '12 at 12:13
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Note, the same could be said about $f\in C^{1}_{0}$. – Patch Jun 10 '12 at 12:14
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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. – Listing Jun 10 '12 at 12:14
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Thank you. I think it also holds for just $f \in C_0^1$. – Misaj Jun 10 '12 at 12:15
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@Patch I think the subscript $0$ denotes that $\lim_{|x|\to \infty} f(x) =0$, rather than compact support. – Ragib Zaman Jun 10 '12 at 12:18
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Oh. I'd never seen that notation before. – Patch Jun 10 '12 at 12:18
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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. – Zachi Evenor Jun 10 '12 at 12:30
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6People 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". – Siminore Jun 10 '12 at 12:40
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1I do not think that this is a duplicate of that question. – davidlowryduda Jun 10 '12 at 12:59
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HINTs
- A continuous function on a compact interval is uniformly continuous.
- $\lim_{|x| \to \infty} f(x) = 0$ means that $\forall \epsilon...$
- Split up the domain to use these two properties.
davidlowryduda
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