# 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, , .

The notion of a zero set as used above is somewhat unusual and does not agree with the zero set of an ordinary function (not a generalized function) as the set of points where that function assumes the value zero. Of course, the statement "fx=0" has no meaning for generalized functions .
A point in the support of a generalized function is called an essential point of , cf. [a4].