# Definition:Support of Continuous Mapping

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## Definition

### 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:

- $\map \supp f = \cl {\set {x \in \R^n: \map f x \ne 0} }$

### General topological group

Let $X$ be a topological space.

Let $G$ be a topological group with identity $e$.

Let $f : X \to G$ be a continuous mapping.

The **support of $f$** is the closure of the set of elements of $X$ that do not map to $e$ under $f$:

- $\map \supp f = \cl {\set {x \in X: \map f x \ne e} }$