In mathematics, the support of a function is the set of points where the function is not zero-valued or, in the case of functions defined on a topological space, the closure of that set. This concept is used very widely in mathematical analysis. In the form of functions with support that is bounded, it also plays a major part in various types of mathematical duality theories.
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.