Support (Mathematics)
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Support is a mathematical concept. It denotes the part of the domain (input set) of a given function where the function takes non-zero values. If f is a function defined on a set X, the support of f is the set of points x where f(x) is non-zero.
The concept is used to distinguish the "active" part of a function from the part where it is zero. Thus support is a tool for defining localized functions and those that vanish everywhere outside a bounded region.
Definition
When the domain is an ordinary set, support is defined as: supp(f) = { x : f(x) ≠ 0 }. However, when the domain is a topological space (a set with a notion of closeness), the closure of that set is often taken instead. This is because the closure yields a closed set, which simplifies certain analytic operations.
Compact support
A function is said to have compact support if its support is a compact set—for example, in ℝ, a closed and bounded set. Functions that vanish outside a bounded region are important in analysis: they form the test functions in distribution theory, and they facilitate differentiation and integration by parts, since boundary terms vanish.
Support in probability
In probability and statistics, the support of a probability distribution is the set of values where the random variable has non-zero probability—for instance, the support of the normal distribution is the entire real line, while the support of the uniform distribution on [0,1] is that interval alone. Support is important when comparing distributions or performing statistical inference.
Applications
Support is used in measure theory, various forms of partial differential equations, Fourier analysis, and computational statistics. For example, bump functions (those with compact support and continuous derivatives) are used to construct partitions of unity and to decompose general functions into localized pieces.