In mathematics, the support of a real-valued function f {\displaystyle f} is the subset of the function's domain consisting of those elements that are not mapped to zero. If the domain of f {\displaystyle f} is a topological space, then the support of f {\displaystyle f} is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.
Formulation Suppose that f : X → R {\displaystyle f:X\to \mathbb {R} } is a real-valued function whose domain is an arbitrary set X . {\displaystyle X.} The set-theoretic support of f , {\displaystyle f,} written supp ( f ) , {\displaystyle \operatorname {supp} (f),} is the set of points in X {\displaystyle X} where f {\displaystyle f} is non-zero:
supp ( f ) = { x ∈ X : f ( x ) ≠ 0 } . {\displaystyle \operatorname {supp} (f)=\{x\in X\,:\,f(x)\neq 0\}.}
The support of f {\displaystyle f} is the smallest subset of X {\displaystyle X} with the property that f {\displaystyle f} is zero on the subset's complement. If f ( x ) = 0 {\displaystyle f(x)=0} for all but a finite number of points x ∈ X , {\displaystyle x\in X,} then f {\displaystyle f} is said to have finite support. If the set X {\displaystyle X} has an additional structure (for example, a topology), then the support of f {\displaystyle f} is defined in an analogous way as the smallest subset of X {\displaystyle X} of an appropriate type such that f {\displaystyle f} vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than R {\displaystyle \mathbb {R} } and to other objects, such as measures or distributions.
Closed support The most common situation occurs when X {\displaystyle X} is a topological space (such as the real line or n {\displaystyle n} -dimensional Euclidean space) and f : X → R {\displaystyle f:X\to \mathbb {R} } is a continuous real- (or complex-) valued function. In this case, the support of f {\displaystyle f} , supp ( f ) {\displaystyle \operatorname {supp} (f)} , or the closed support of f {\displaystyle f} , is defined topologically as the closure (taken in X {\displaystyle X} ) of the subset of X {\displaystyle X} where f {\displaystyle f} is non-zero that is,
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