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A function defined in some domain of  $  E  ^ {n} $, 
having compact support belonging to this domain. More precisely, suppose that the function  $  f ( x) = f ( x _ {1} \dots x _ {n} ) $
is defined on a domain  $  \Omega \subset  E  ^ {n} $. 
The support of  $  f $
is the closure of the set of points  $  x \in \Omega $
for which  $  f ( x) $
is different from zero  $  ( f ( x) \neq 0) $. 
Thus one can also say that a function of compact support in  $  \Omega $
is a function defined on  $  \Omega $
such that its support  $  \Lambda $
is a closed bounded set located at a distance from the boundary  $  \Gamma $
of  $  \Omega $
by a number greater than  $  \delta > 0 $, 
where  $  \delta $
is sufficiently small.

One usually considers  $  k $-
times continuously-differentiable functions of compact support, where  $  k $
is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function

$$ 
\psi ( x)  = \ 
\left \{

\begin{array}{ll}
e ^ {- 1/( | x |  ^ {2} - 1) } ,  & | x | < 1,
 | x |  ^ {2} = \sum _ {j = 1 } ^ { n }  x _ {j}  ^ {2} ,  \\
0 ,  & | x | \geq  1 ,  \\
\end{array}

 \right .$$

can serve as an example of an infinitely-differentiable function of compact support in a domain  $  \Omega $
containing the sphere  $  | x | \leq  1 $.

The set of all infinitely-differentiable functions of compact support in a domain  $  \Omega \subset  E  ^ {n} $
is denoted by  $  D $. 
On  $  D $
one can define linear functionals (generalized functions, cf. [[Generalized function|Generalized function]]). With the aid of functions  $  v \in D $
one can define generalized solutions (cf. [[Generalized solution|Generalized solution]]) of boundary value problems.

In theorems concerned with problems on finding generalized solutions, it is often important to know whether  $  D $
is dense in some concrete space of functions. It is known, for example, that if the boundary  $  \Gamma $
of a bounded domain  $  \Omega \subset  E  ^ {n} $
is sufficiently smooth, then  $  D $
is dense in the space of functions

$$ 
{W ^ { o }  } {} _ {p}  ^ {r} ( \Omega )  = \ 
\left \{ {f } : {
f \in W _ {p}  ^ {r} ( \Omega ),\ 
\left . 
\frac{\partial   ^ {s} f }{\partial  n  ^ {s} }
 \right | _  \Gamma  = 0,\ 
s = 0 \dots r - 1
 } \right \}
$$

( $  1 \leq  p \leq  \infty $), 
that is, in the [[Sobolev space|Sobolev space]] of functions of class  $  W _ {p}  ^ {r} ( \Omega ) $
that vanish on  $  \Gamma $
along with their normal derivatives of order up to and including  $  r - 1 $(
$  r = 1, 2 ,\dots $).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1950–1951)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Lions,   E. Magenes,   "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR></table>