Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The support of a function is defined in Wikipedia as "the set of points where the function is not zero-valued, or the closure of that set".

Functions with compact support in $X$ are defined in Wikipedia as "those with support that is a compact subset of $X$. For example, if $X$ is the real line, they are functions of bounded support and therefore vanish at infinity (and negative infinity)".

Why functions vanishing at infinity are considered as having compact support?

An example of a function vanishing at infinity is $f(x) = \frac{1}{x^2+1}$, it's support is $\mathbb{R}$.

The compactness of a subset $K$ is defined as "every arbitrary collection of open subsets of $X$ such that covers $K$, there is a finite subset also covers $K$".

Now $\mathbb{R}$ is not compact, we can't say $f(x) = \frac{1}{x^2+1}$ has a compact support, am I right there?

share|improve this question

2 Answers 2

up vote 11 down vote accepted

Every function with compact support vanishes at infinity; this is what the Wikipedia article states. The converse is not true, as illustrated by your example.

share|improve this answer
thanks. glad to know my math intuition is right, sad to learn my english understanding is poor :p –  athos Oct 2 '13 at 3:37

You have the implication the wrong way round: $f$ compactly supported implies that $f$ vanishes at infinity. Vanishing at infinity is a necessary but not sufficient condition for a function to be compactly supported.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.