# why Functions with compact support include those vanish at infinity?

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

Compact 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?

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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.