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