Compact support

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.

Suppose that f : X → R is a real-valued function whose domain is an arbitrary set X. The set-theoretic support of f, written supp(f), is the set of points in X where f is non-zero

The support of f is the smallest subset of X with the property that f is zero on the subset's complement, meaning that the non-zero values of f "live" on supp(f). If f(x) = 0 for all but a finite number of points x in X, then f is said to have finite support.

If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that 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 and to other objects, such as measures or distributions.

The most common situation occurs when X is a topological space (such as the real line or n-dimensional Euclidean space) and f : X → R is a continuous real (or complex)-valued function. In this case, the support of f is defined topologically as the closure of the subset of X where f is non-zero i.e.,

Since the intersection of closed sets is closed, supp(f) is the intersection of all closed sets that contain the set-theoretic support of f.

For example, if f : R → R is the function defined by

then the support of f is the closed interval [−1,1], since f is non-zero on the open interval (−1,1) and the closure of this set is [−1,1].

The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that f : X → R (or C) be continuous.

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