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I am looking for a term that describes function that is "localized". What I mean is a function that is non zero in a bounded range and zero else where, such as the a rectangle pulse function. But would also like that the term cover functions such as gaussian that approach zero as you get away from the peak. If you have the whole definition with a reference would be wonderful. I couldn't find anything that pleased me in a basic googling.

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If it is constant in the non-zero range, you can call it a step function. – Aryabhata Jan 17 '11 at 22:52
A "bump function" has additional constraints (smoothness), but may in practice be what you're looking for. – Peter Taylor Jan 17 '11 at 22:58
Depends on your application, in addition to "vanishing at infinity" as described by Qiaochu and Arturo below, sometimes it is also useful to specify exactly how quick it vanishes: logarithmically? at a power law? faster than any power law? This allows you to rule in functions that do look like they have bumps (the Gaussian for example) while ruling out functions that look more spread-out (something like $1/(1 +|x|^2)$. – Willie Wong Jan 18 '11 at 0:10
In this question, the answerer uses the term "locally constant":… – PEV Jan 20 '11 at 1:39
up vote 6 down vote accepted

Given a function $f\colon X\to \mathbb{R}$, the support of $f$ is defined to be $$\mathrm{sup}(f) = \{x\in X\mid f(x)\neq 0\}.$$

So in the first part, you are talking about "real variable functions of bounded support". The "real variable" can be dropped if the domain is understood.

For functions that have limit equal to $0$ at infinity you talk about functions that "vanish at $\infty$". (Thanks to Qiaochu for mentioning this, as I missed it).

Note that any function with bounded support will necessarily vanish at infinity, but $y = e^{-x^2}$ shows you can have functions that vanish at $\infty$ but are not of bounded support.

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The second term you're looking for is vanishing at infinity. At least on locally compact spaces a natural generalization of the first term is compact support.

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