Editing Support (mathematics) (section)

==Closed support==

The most common situation occurs when <math>X</math> is a [[topological space]] (such as the [[real line]] or <math>n</math>-dimensional [[Euclidean space]]) and <math>f : X \to \R</math> is a [[Continuous function|continuous]] real- (or [[Complex number|complex]]-) valued function. In this case, the '''{{em|{{visible anchor|support}}}} of <math>f</math>''', <math>\operatorname{supp}(f)</math>, or the '''{{em|{{visible anchor|closed support}}}}''' '''of''' '''<math>f</math>''', is defined topologically as the [[Closure (topology)|closure]] (taken in <math>X</math>) of the subset of <math>X</math> where <math>f</math> is non-zero<ref name='folland'>{{cite book|last=Folland|first=Gerald B.|year=1999|title=Real Analysis, 2nd ed.|page=132|location=New York|publisher=John Wiley}}</ref><ref name='hormander'>{{cite book|last=Hörmander|first=Lars|year=1990|title=Linear Partial Differential Equations I, 2nd ed.|page=14|location=Berlin|publisher=Springer-Verlag}}</ref><ref name=Pasc>{{cite book|last=Pascucci|first=Andrea|year=2011|title=PDE and Martingale Methods in Option Pricing|page=678|isbn=978-88-470-1780-1|doi=10.1007/978-88-470-1781-8|location=Berlin|publisher=Springer-Verlag|series=Bocconi & Springer Series}}</ref> that is,
<math display="block">\operatorname{supp}(f) := \operatorname{cl}_X\left(\{x \in X \,:\, f(x) \neq 0 \}\right) = \overline{f^{-1}\left(\{ 0 \}^{\mathrm{c}}\right)}.</math>Since the intersection of closed sets is closed, <math>\operatorname{supp}(f)</math> is the intersection of all closed sets that contain the set-theoretic support of <math>f.</math> Note that if the function <math>f: \mathbb{R}^n \supseteq X \to \mathbb{R}</math> is defined on an open subset <math>X \subseteq \mathbb{R}^n</math>, then the closure is still taken with respect to <math>X</math> and not with respect to the ambient <math>\mathbb{R}^n</math>.

For example, if <math>f : \R \to \R</math> is the function defined by
<math display="block">f(x) = \begin{cases} 1 - x^2 & \text{if } |x| < 1 \\ 0 & \text{if } |x| \geq 1 \end{cases}</math>
then <math>\operatorname{supp}(f)</math>, the support of <math>f</math>, or the closed support of <math>f</math>, is the closed interval <math>[-1, 1],</math> since <math>f</math> is non-zero on the open interval <math>(-1, 1)</math> and the [[Closure (topology)|closure]] of this set is <math>[-1, 1].</math>

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 <math>f : X \to \R</math> (or <math>f : X \to \Complex</math>) be continuous.<ref>{{cite book|last=Rudin|first=Walter|year=1987|title=Real and Complex Analysis, 3rd ed.|page=38|location=New York|publisher=McGraw-Hill}}</ref>