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Inmathematics, thesupport of areal-valuedfunction${\displaystyle f}$ is thesubset of the function'sdomain consisting of those elements that are not mapped to zero. If the domain of${\displaystyle f}$ is atopological space, then the support of${\displaystyle f}$ is instead defined as the smallestclosed set containing all points not mapped to zero. This concept is used widely inmathematical analysis.

Suppose that${\displaystyle f:X\to \mathbb{R}}$ is a real-valued function whosedomain is an arbitrary set${\displaystyle X.}$ Theset-theoretic support of${\displaystyle f,}$ written${\displaystyle \mathrm{supp}(f),}$ is the set of points in${\displaystyle X}$ where${\displaystyle f}$ is non-zero:$${\displaystyle \mathrm{supp}(f)=\{x\in X:f(x)\ne 0\}.}$$

The support of${\displaystyle f}$ is the smallest subset of${\displaystyle X}$ with the property that${\displaystyle f}$ is zero on the subset's complement. If${\displaystyle f(x)=0}$ for all but a finite number of points${\displaystyle x\in X,}$ then${\displaystyle f}$ is said to havefinite support.

If the set${\displaystyle X}$ has an additional structure (for example, atopology), then the support of${\displaystyle f}$ is defined in an analogous way as the smallest subset of${\displaystyle X}$ of an appropriate type such that${\displaystyle f}$ 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${\displaystyle \mathbb{R}}$ and to other objects, such asmeasures ordistributions.

The most common situation occurs when${\displaystyle X}$ is atopological space (such as thereal line or${\displaystyle n}$-dimensionalEuclidean space) and${\displaystyle f:X\to \mathbb{R}}$ is acontinuous real- (orcomplex-) valued function. In this case, thesupport of${\displaystyle f}$,${\displaystyle \mathrm{supp}(f)}$, or theclosed supportof${\displaystyle f}$, is defined topologically as theclosure (taken in${\displaystyle X}$) of the subset of${\displaystyle X}$ where${\displaystyle f}$ is non-zero^[1][2][3] that is,$${\displaystyle \mathrm{supp}(f):={\mathrm{cl}}_X\left(\{x\in X:f(x)\ne 0\}\right)=\overline{f^{-1}\left(\{0{\}}^{\mathrm{c}}\right)}.}$$Since the intersection of closed sets is closed,${\displaystyle \mathrm{supp}(f)}$ is the intersection of all closed sets that contain the set-theoretic support of${\displaystyle f.}$ Note that if the function${\displaystyle f:{\mathbb{R}}^n\supseteq X\to \mathbb{R}}$ is defined on an open subset${\displaystyle X\subseteq {\mathbb{R}}^n}$, then the closure is still taken with respect to${\displaystyle X}$ and not with respect to the ambient${\displaystyle {\mathbb{R}}^n}$.

For example, if${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ is the function defined bythen${\displaystyle \mathrm{supp}(f)}$, the support of${\displaystyle f}$, or the closed support of${\displaystyle f}$, is the closed interval${\displaystyle [-1,1],}$ since${\displaystyle f}$ is non-zero on the open interval${\displaystyle (-1,1)}$ and theclosure of this set is${\displaystyle [-1,1].}$

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${\displaystyle f:X\to \mathbb{R}}$ (or${\displaystyle f:X\to \mathbb{C}}$) be continuous.^[4]

Functions withcompact support on a topological space${\displaystyle X}$ are those whose closed support is acompact subset of${\displaystyle X.}$ If${\displaystyle X}$ is the real line, or${\displaystyle n}$-dimensional Euclidean space, then a function has compact support if and only if it hasbounded support, since a subset of${\displaystyle {\mathbb{R}}^n}$ is compact if and only if it is closed and bounded.

For example, the function${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ defined above is a continuous function with compact support${\displaystyle [-1,1].}$ If${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ is a smooth function then because${\displaystyle f}$ is identically${\displaystyle 0}$ on the open subset${\displaystyle {\mathbb{R}}^n\setminus \mathrm{supp}(f),}$ all of${\displaystyle f}$'s partial derivatives of all orders are also identically${\displaystyle 0}$ on${\displaystyle {\mathbb{R}}^n\setminus \mathrm{supp}(f).}$

The condition of compact support is stronger than the condition ofvanishing at infinity. For example, the function${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ defined by$${\displaystyle f(x)=\frac{1}{1+x^2}}$$vanishes at infinity, since${\displaystyle f(x)\to 0}$ as${\displaystyle |x|\to \mathrm{\infty },}$ but its support${\displaystyle \mathbb{R}}$ is not compact.

Real-valued compactly supportedsmooth functions on aEuclidean space are calledbump functions.Mollifiers are an important special case of bump functions as they can be used indistribution theory to createsequences of smooth functions approximating nonsmooth (generalized) functions, viaconvolution.

Ingood cases, functions with compact support aredense 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 oflimits, for any${\displaystyle \epsilon >0,}$ any function${\displaystyle f}$ on the real line${\displaystyle \mathbb{R}}$ that vanishes at infinity can be approximated by choosing an appropriate compact subset${\displaystyle C}$ of${\displaystyle \mathbb{R}}$ such thatfor all${\displaystyle x\in X,}$ where${\displaystyle I_C}$ is theindicator function of${\displaystyle C.}$ Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.

If${\displaystyle X}$ is a topologicalmeasure space with aBorel measure${\displaystyle \mu }$ (such as${\displaystyle {\mathbb{R}}^n,}$ or aLebesgue measurable subset of${\displaystyle {\mathbb{R}}^n,}$ equipped with Lebesgue measure), then one typically identifies functions that are equal${\displaystyle \mu }$-almost everywhere. In that case, theessential support of a measurable function${\displaystyle f:X\to \mathbb{R}}$ written${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f),}$ is defined to be the smallest closed subset${\displaystyle F}$ of${\displaystyle X}$ such that${\displaystyle f=0}$${\displaystyle \mu }$-almost everywhere outside${\displaystyle F.}$ Equivalently,${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)}$ is the complement of the largestopen set on which${\displaystyle f=0}$${\displaystyle \mu }$-almost everywhere^[5]$${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f):=X\setminus \bigcup \left\{\mathrm{\Omega }\subseteq X:\mathrm{\Omega }\text{ is open and }f=0\mu \text{-almost everywhere in }\mathrm{\Omega }\right\}.}$$

The essential support of a function${\displaystyle f}$ depends on themeasure${\displaystyle \mu }$ as well as on${\displaystyle f,}$ and it may be strictly smaller than the closed support. For example, if${\displaystyle f:[0,1]\to \mathbb{R}}$ is theDirichlet function that is${\displaystyle 0}$ on irrational numbers and${\displaystyle 1}$ on rational numbers, and${\displaystyle [0,1]}$ is equipped with Lebesgue measure, then the support of${\displaystyle f}$ is the entire interval${\displaystyle [0,1],}$ but the essential support of${\displaystyle f}$ is empty, since${\displaystyle f}$ 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${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)}$ is often written simply as${\displaystyle \mathrm{supp}(f)}$ and referred to as the support.^[5][6]

If${\displaystyle M}$ is an arbitrary set containing zero, the concept of support is immediately generalizable to functions${\displaystyle f:X\to M.}$ Support may also be defined for anyalgebraic structure withidentity (such as agroup,monoid, orcomposition algebra), in which the identity element assumes the role of zero. For instance, the family${\displaystyle {\mathbb{Z}}^{\mathbb{N}}}$ of functions from thenatural numbers to theintegers is theuncountable set of integer sequences. The subfamily${\displaystyle \left\{f\in {\mathbb{Z}}^{\mathbb{N}}:f\text{ has finite support }\right\}}$ 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 asgroup rings andfree abelian groups.^[7]

Inprobability theory, the support of aprobability distribution can be loosely thought of as theclosure 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 asigma algebra, rather than on a topological space.

More formally, if${\displaystyle X:\mathrm{\Omega }\to \mathbb{R}}$ is a random variable on${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$ then the support of${\displaystyle X}$ is the smallest closed set${\displaystyle R_X\subseteq \mathbb{R}}$ such that${\displaystyle P\left(X\in R_X\right)=1.}$

In practice however, the support of adiscrete random variable${\displaystyle X}$ is often defined as the set${\displaystyle R_X=\{x\in \mathbb{R}:P(X=x)>0\}}$ and the support of acontinuous random variable${\displaystyle X}$ is defined as the set${\displaystyle R_X=\{x\in \mathbb{R}:f_X(x)>0\}}$ where${\displaystyle f_X(x)}$ is aprobability density function of${\displaystyle X}$ (theset-theoretic support).^[8]

Note that the wordsupport can refer to thelogarithm of thelikelihood of a probability density function.^[9]

It is possible also to talk about the support of adistribution, such as theDirac delta function${\displaystyle \delta (x)}$ on the real line. In that example, we can consider test functions${\displaystyle F,}$ which aresmooth functions with support not including the point${\displaystyle 0.}$ Since${\displaystyle \delta (F)}$ (the distribution${\displaystyle \delta }$ applied aslinear functional to${\displaystyle F}$) is${\displaystyle 0}$ for such functions, we can say that the support of${\displaystyle \delta }$ is${\displaystyle \{0\}}$ only. Since measures (includingprobability 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${\displaystyle f}$ is a distribution, and that${\displaystyle U}$ is an open set in Euclidean space such that, for all test functions${\displaystyle \varphi }$ such that the support of${\displaystyle \varphi }$ is contained in${\displaystyle U,}$${\displaystyle f(\varphi )=0.}$ Then${\displaystyle f}$ is said to vanish on${\displaystyle U.}$ Now, if${\displaystyle f}$ vanishes on an arbitrary family${\displaystyle U_{\alpha }}$ of open sets, then for any test function${\displaystyle \varphi }$ supported in$\bigcup U_{\alpha },$ a simple argument based on the compactness of the support of${\displaystyle \varphi }$ and a partition of unity shows that${\displaystyle f(\varphi )=0}$ as well. Hence we can define thesupport of${\displaystyle f}$ as the complement of the largest open set on which${\displaystyle f}$ vanishes. For example, the support of the Dirac delta is${\displaystyle \{0\}.}$

InFourier analysis in particular, it is interesting to study thesingular support of a distribution. This has the intuitive interpretation as the set of points at which a distributionfails to be a smooth function.

For example, theFourier transform of theHeaviside step function can, up to constant factors, be considered to be${\displaystyle 1/x}$ (a function)except at${\displaystyle x=0.}$ While${\displaystyle x=0}$ is clearly a special point, it is more precise to say that the transform of the distribution has singular support${\displaystyle \{0\}}$: it cannot accurately be expressed as a function in relation to test functions with support including${\displaystyle 0.}$ Itcan be expressed as an application of aCauchy principal valueimproper integral.

For distributions in several variables, singular supports allow one to definewave front sets and understandHuygens' principle in terms ofmathematical analysis. 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).

An abstract notion offamily of supports on atopological space${\displaystyle X,}$ suitable forsheaf theory, was defined byHenri Cartan. In extendingPoincaré duality tomanifolds that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for exampleAlexander–Spanier cohomology.

Bredon,Sheaf Theory (2nd edition, 1997) gives these definitions. A family${\displaystyle \mathrm{\Phi }}$ of closed subsets of${\displaystyle X}$ is afamily of supports, if it isdown-closed and closed underfinite union. Itsextent is the union over${\displaystyle \mathrm{\Phi }.}$ Aparacompactifying family of supports that satisfies further that any${\displaystyle Y}$ in${\displaystyle \mathrm{\Phi }}$ is, with thesubspace topology, aparacompact space; and has some${\displaystyle Z}$ in${\displaystyle \mathrm{\Phi }}$ which is aneighbourhood. If${\displaystyle X}$ is alocally compact space, assumedHausdorff, the family of allcompact subsets satisfies the further conditions, making it paracompactifying.

• Bounded function – Mathematical function whose set of values is bounded
• Bump function – Smooth and compactly supported function
• Support of a module
• Titchmarsh convolution theorem

• Rudin, Walter (1991).Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 978-0-07-054236-5.OCLC 21163277.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.

• Set theory
• Real analysis
• Topology
• Topology of function spaces
• Schwartz distributions

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