# Support of a function

From Encyclopedia of Mathematics

2020 Mathematics Subject Classification: *Primary:* 54A [MSN][ZBL]

Let $X$ be a topological space and $f:X\to \mathbb R$ a function. The support of $f$, denoted by ${\rm supp}\, (f)$ is the smallest closed set outside of which the function $f$ vanishes identically. ${\rm supp}\, (f)$ can also be characterized as

- the complent of the union of all sets on which $f$ vanishes identically
- the closure of the set $\{f\neq 0\}$.

The same concept can be readily extended to maps taking values in a vector space or more generally in an additive group.

A function $f$ is said to have compact support if ${\rm supp}\, (f)$ is compact. If the target $V$ is a vector space, the set of functions $f:X\to V$ with compact support is also a vector space.

#### References

[Ru] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 38\, . |

**How to Cite This Entry:**

Support of a function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_function&oldid=28948

This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article