Definition:Support

Real-Valued Function on an Abstract Set
Let $f: S \to \R$ be a real-valued function.

The support of $f$ is the set of elements $x$ of $S$ whose values under $f$ are non-zero.

That is:
 * $\operatorname{supp} \left({f}\right) = \left\{{x \in S: f \left({x}\right) \ne 0}\right\}$

Continuous Real-Valued Function in $\R^n$
Let $f: \R^n \to \R$ be a continuous real-valued function.

The support of $f$ is the closure of the set of elements $x$ of $\R^n$ whose values under $f$ are non-zero.

That is:
 * $\operatorname{supp} \left({f}\right) = \overline{\left\{{x \in \R^n: f \left({x}\right) \ne 0}\right\}}$

Distribution
Let $\Omega \subseteq \R^n$ be an open set.

Let $\mathcal D\left( \Omega \right)$ be the space of continuous functions compactly supported in $\Omega$.

Let $T \in \mathcal D' \left( \Omega \right)$ be a distribution.

The support $\mathrm{supp}\;T \subseteq \Omega$ of $T$ is defined by: $\displaystyle x \notin \operatorname{supp} T$ if and only if there exists an open neighborhood $U$ of $x$ such that for all $\phi \in \mathcal D\left( \Omega \right)$ such that $\mathrm{supp}\;\phi \subseteq U$ we have $T\left(\phi\right) = 0$.