# Support of a generalized function

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The set of those (and only those) points such that in any neighbourhood of them the generalized function does not vanish. A generalized function in vanishes in an open set if for all . Using a partition of unity it can be proved that if a generalized function in vanishes in some neighbourhood for each point , then vanishes in . The union of all neighbourhoods in which vanishes is called the zero set of and is denoted by . The support of , denoted by , is the complement of in , that is, is a closed set in . If is a continuous function in , then an equivalent definition of the support of is the following: is the closure in of the complement of the set of points at which vanishes (cf. Support of a function). For example, , .

The singular support () of a generalized function is the set of those (and only those) points such that in any neighbourhood of them the generalized function is not equal to a -function. For example, , .