In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, the support of a real-valued
In mathematics, value may refer to several, strongly related notions.
In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an in ...

function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
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$f$ is the subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

containing the elements which are not mapped to zero. If the domain of $f$ is a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, the support of $f$ is instead defined as the smallest closed set
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

containing all points not mapped to zero. This concept is used very widely in mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

.
Formulation

Suppose that $f\; :\; X\; \backslash to\; \backslash R$ is a real-valued function whosedomain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

is an arbitrary set $X.$ The of $f,$ written $\backslash operatorname(f),$ is the set of points in $X$ where $f$ is non-zero:
$$\backslash operatorname(f)\; =\; \backslash .$$
The support of $f$ is the smallest subset of $X$ with the property that $f$ is zero on the subset's complement. If $f(x)\; =\; 0$ for all but a finite number of points $x\; \backslash in\; X,$ then $f$ is said to have .
If the set $X$ has an additional structure (for example, a topology), then the support of $f$ is defined in an analogous way as the smallest subset of $X$ of an appropriate type such that $f$ vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than $\backslash R$ and to other objects, such as measures or distributions.
Closed support

The most common situation occurs when $X$ is atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

(such as the real line
In mathematics
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or $n$-dimensional Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

) and $f\; :\; X\; \backslash to\; \backslash R$ is a continuous
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Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

real (or complex
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)-valued function. In this case, the of $f$ is defined topologically as the closure (taken in $X$) of the subset of $X$ where $f$ is non-zero that is,
$$\backslash operatorname(f)\; :=\; \backslash operatorname\_X\backslash left(\backslash \backslash right)\; =\; \backslash overline.$$
Since the intersection of closed sets is closed, $\backslash operatorname(f)$ is the intersection of all closed sets that contain the set-theoretic support of $f.$
For example, if $f\; :\; \backslash R\; \backslash to\; \backslash R$ is the function defined by
$$f(x)\; =\; \backslash begin\; 1\; -\; x^2\; \&\; \backslash text\; ,\; x,\; <\; 1\; \backslash \backslash \; 0\; \&\; \backslash text\; ,\; x,\; \backslash geq\; 1\; \backslash end$$
then the support of $f$ is the closed interval $;\; href="/html/ALL/s/1,\_1.html"\; ;"title="1,\; 1">1,\; 1$ since $f$ is non-zero on the open interval $(-1,\; 1)$ and the closure of this set is $;\; href="/html/ALL/s/1,\_1.html"\; ;"title="1,\; 1">1,\; 1$
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 $f\; :\; X\; \backslash to\; \backslash R$ (or $f\; :\; X\; \backslash to\; \backslash Complex$) be continuous.
Compact support

Functions with on a topological space $X$ are those whose closed support is acompact
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subset of $X.$ If $X$ is the real line, or $n$-dimensional Euclidean space, then a function has compact support if and only if it has , since a subset of $\backslash R^n$ is compact if and only if it is closed and bounded.
For example, the function $f\; :\; \backslash R\; \backslash to\; \backslash R$ defined above is a continuous function with compact support $;\; href="/html/ALL/s/1,\_1.html"\; ;"title="1,\; 1">1,\; 1$
If $f\; :\; \backslash R^n\; \backslash to\; \backslash R$ is a smooth function then because $f$ is identically $0$ on the open subset $\backslash R^n\; \backslash setminus\; \backslash operatorname(f),$ all of $f$'s partial derivatives of all orders are also identically $0$ on $\backslash R^n\; \backslash setminus\; \backslash operatorname(f).$
The condition of compact support is stronger than the condition of vanishing at infinity. For example, the function $f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by
$$f(x)\; =\; \backslash frac$$
vanishes at infinity, since $f(x)\; \backslash to\; 0$ as $,\; x,\; \backslash to\; \backslash infty,$ but its support $\backslash R$ is not compact.
Real-valued compactly supported smooth functions on a Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

are called bump functions. Mollifiers are an important special case of bump functions as they can be used in Distribution (mathematics), distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.
In Well-behaved, good cases, functions with compact support are Dense set, dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of Limit (mathematics), limits, for any $\backslash varepsilon\; >\; 0,$ any function $f$ on the real line $\backslash R$ that vanishes at infinity can be approximated by choosing an appropriate compact subset $C$ of $\backslash R$ such that
$$\backslash left,\; f(x)\; -\; I\_C(x)\; f(x)\backslash \; <\; \backslash varepsilon$$
for all $x\; \backslash in\; X,$ where $I\_C$ is the indicator function of $C.$ Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.
Essential support

If $X$ is a topological measure space with a Borel measure $\backslash mu$ (such as $\backslash R^n,$ or a Lebesgue measure, Lebesgue measurable subset of $\backslash R^n,$ equipped with Lebesgue measure), then one typically identifies functions that are equal $\backslash mu$-almost everywhere. In that case, the of a measurable function $f\; :\; X\; \backslash to\; \backslash R$ written $\backslash operatorname(f),$ is defined to be the smallest closed subset $F$ of $X$ such that $f\; =\; 0$ $\backslash mu$-almost everywhere outside $F.$ Equivalently, $\backslash operatorname(f)$ is the complement of the largest open set on which $f\; =\; 0$ $\backslash mu$-almost everywhere $$\backslash operatorname(f)\; :=\; X\; \backslash setminus\; \backslash bigcup\; \backslash left\backslash .$$ The essential support of a function $f$ depends on the Measure (mathematics), measure $\backslash mu$ as well as on $f,$ and it may be strictly smaller than the closed support. For example, if $f\; :\; [0,\; 1]\; \backslash to\; \backslash R$ is the Dirichlet function that is $0$ on irrational numbers and $1$ on rational numbers, and $[0,\; 1]$ is equipped with Lebesgue measure, then the support of $f$ is the entire interval $[0,\; 1],$ but the essential support of $f$ is empty, since $f$ is equal almost everywhere to the zero function. In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so $\backslash operatorname(f)$ is often written simply as $\backslash operatorname(f)$ and referred to as the support.Generalization

If $M$ is an arbitrary set containing zero, the concept of support is immediately generalizable to functions $f\; :\; X\; \backslash to\; M.$ Support may also be defined for any algebraic structure with Identity element, identity (such as a Group (mathematics), group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family $\backslash Z^$ of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily $\backslash left\backslash $ is the countable set of all integer sequences that have only finitely many nonzero entries. Functions of finite support are used in defining algebraic structures such as Group ring, group rings and Free abelian group, free abelian groups.In probability and measure theory

In probability theory, the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution. There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra, rather than on a topological space. More formally, if $X\; :\; \backslash Omega\; \backslash to\; \backslash R$ is a random variable on $(\backslash Omega,\; \backslash mathcal,\; P)$ then the support of $X$ is the smallest closed set $R\_X\; \backslash subseteq\; \backslash R$ such that $P\backslash left(X\; \backslash in\; R\_X\backslash right)\; =\; 1.$ In practice however, the support of a discrete random variable $X$ is often defined as the set $R\_X\; =\; \backslash $ and the support of a continuous random variable $X$ is defined as the set $R\_X\; =\; \backslash $ where $f\_X(x)$ is a probability density function of $X$ (the #set-theoretic support, set-theoretic support). Note that the word can refer to the logarithm of the likelihood function, likelihood of a probability density function.Support of a distribution

It is possible also to talk about the support of a Distribution (mathematics), distribution, such as the Dirac delta function $\backslash delta(x)$ on the real line. In that example, we can consider test functions $F,$ which are smooth functions with support not including the point $0.$ Since $\backslash delta(F)$ (the distribution $\backslash delta$ applied as linear functional to $F$) is $0$ for such functions, we can say that the support of $\backslash delta$ is $\backslash $ only. Since measures (including probability measures) on the real line are special cases of distributions, we can also speak of the support of a measure in the same way. Suppose that $f$ is a distribution, and that $U$ is an open set in Euclidean space such that, for all test functions $\backslash phi$ such that the support of $\backslash phi$ is contained in $U,$ $f(\backslash phi)\; =\; 0.$ Then $f$ is said to vanish on $U.$ Now, if $f$ vanishes on an arbitrary family $U\_$ of open sets, then for any test function $\backslash phi$ supported in $\backslash bigcup\; U\_,$ a simple argument based on the compactness of the support of $\backslash phi$ and a partition of unity shows that $f(\backslash phi)\; =\; 0$ as well. Hence we can define the of $f$ as the complement of the largest open set on which $f$ vanishes. For example, the support of the Dirac delta is $\backslash .$Singular support

In Fourier analysis in particular, it is interesting to study the of a distribution. This has the intuitive interpretation as the set of points at which a distribution . For example, the Fourier transform of the Heaviside step function can, up to constant factors, be considered to be $1/x$ (a function) at $x\; =\; 0.$ While $x\; =\; 0$ is clearly a special point, it is more precise to say that the transform of the distribution has singular support $\backslash $: it cannot accurately be expressed as a function in relation to test functions with support including $0.$ It be expressed as an application of a Cauchy principal value integral. For distributions in several variables, singular supports allow one to define and understand Huygens' principle in terms ofmathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).
Family of supports

An abstract notion of on atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

$X,$ suitable for sheaf theory, was defined by Henri Cartan. In extending Poincaré duality to manifolds that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example Alexander–Spanier cohomology.
Bredon, ''Sheaf Theory'' (2nd edition, 1997) gives these definitions. A family $\backslash Phi$ of closed subsets of $X$ is a , if it is down-closed and closed under finite union. Its is the union over $\backslash Phi.$ A family of supports that satisfies further that any $Y$ in $\backslash Phi$ is, with the subspace topology, a paracompact space; and has some $Z$ in $\backslash Phi$ which is a Neighbourhood (topology), neighbourhood. If $X$ is a locally compact space, assumed Hausdorff space, Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.
See also

* * * *Citations

References

* * Set theory Real analysis Topology Topology of function spaces