# Support (mathematics)

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Inputs for which a function's value is non-zero

For other uses in mathematics, see [Support § Mathematics](/source/Support_(disambiguation)#Mathematics).

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In [mathematics](/source/Mathematics), the **support** of a [real-valued](/source/Real_number) [function](/source/Function_(mathematics)) f {\displaystyle f} is the [subset](/source/Subset) of the function's [domain](/source/Domain_of_a_function) consisting of those elements that are not mapped to zero. If the domain of f {\displaystyle f} is a [topological space](/source/Topological_space), then the support of f {\displaystyle f} is instead defined as the smallest [closed set](/source/Closed_set) containing all points not mapped to zero. This concept is used widely in [mathematical analysis](/source/Mathematical_analysis).

## Formulation

Suppose that f : X → R {\displaystyle f:X\to \mathbb {R} } is a real-valued function whose [domain](/source/Domain_of_a_function) is an arbitrary set X . {\displaystyle X.} The ***set-theoretic support*** of f , {\displaystyle f,} written supp ⁡ ( f ) , {\displaystyle \operatorname {supp} (f),} is the set of points in X {\displaystyle X} where f {\displaystyle f} is non-zero: supp ⁡ ( f ) = { x ∈ X : f ( x ) ≠ 0 } . {\displaystyle \operatorname {supp} (f)=\{x\in X\,:\,f(x)\neq 0\}.}

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

If the set X {\displaystyle X} has an additional structure (for example, a [topology](/source/Topology_(structure))), then the support of f {\displaystyle f} is defined in an analogous way as the smallest subset of X {\displaystyle X} of an appropriate type such that f {\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 R {\displaystyle \mathbb {R} } and to other objects, such as [measures](/source/Measure_(mathematics)) or [distributions](/source/Distribution_(mathematics)).

## Closed support

The most common situation occurs when X {\displaystyle X} is a [topological space](/source/Topological_space) (such as the [real line](/source/Real_line) or n {\displaystyle n} -dimensional [Euclidean space](/source/Euclidean_space)) and f : X → R {\displaystyle f:X\to \mathbb {R} } is a [continuous](/source/Continuous_function) real- (or [complex](/source/Complex_number)-) valued function. In this case, the ***support* of f {\displaystyle f}**, supp ⁡ ( f ) {\displaystyle \operatorname {supp} (f)} , or the ***closed support*** **of** **f {\displaystyle f}**, is defined topologically as the [closure](/source/Closure_(topology)) (taken in X {\displaystyle X} ) of the subset of X {\displaystyle X} where f {\displaystyle f} is non-zero[1][2][3] that is, supp ⁡ ( f ) := cl X ⁡ ( { x ∈ X : f ( x ) ≠ 0 } ) = f − 1 ( { 0 } c ) ¯ . {\displaystyle \operatorname {supp} (f):=\operatorname {cl} _{X}\left(\{x\in X\,:\,f(x)\neq 0\}\right)={\overline {f^{-1}\left(\{0\}^{\mathrm {c} }\right)}}.} Since the intersection of closed sets is closed, supp ⁡ ( f ) {\displaystyle \operatorname {supp} (f)} is the intersection of all closed sets that contain the set-theoretic support of f . {\displaystyle f.} Note that if the function f : R n ⊇ X → R {\displaystyle f:\mathbb {R} ^{n}\supseteq X\to \mathbb {R} } is defined on an open subset X ⊆ R n {\displaystyle X\subseteq \mathbb {R} ^{n}} , then the closure is still taken with respect to X {\displaystyle X} and not with respect to the ambient R n {\displaystyle \mathbb {R} ^{n}} .

For example, if f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is the function defined by f ( x ) = { 1 − x 2 if | x | < 1 0 if | x | ≥ 1 {\displaystyle f(x)={\begin{cases}1-x^{2}&{\text{if }}|x|<1\\0&{\text{if }}|x|\geq 1\end{cases}}} then supp ⁡ ( f ) {\displaystyle \operatorname {supp} (f)} , the support of f {\displaystyle f} , or the closed support of f {\displaystyle f} , is the closed interval [ − 1 , 1 ] , {\displaystyle [-1,1],} since f {\displaystyle f} is non-zero on the open interval ( − 1 , 1 ) {\displaystyle (-1,1)} and the [closure](/source/Closure_(topology)) of this set is [ − 1 , 1 ] . {\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 f : X → R {\displaystyle f:X\to \mathbb {R} } (or f : X → C {\displaystyle f:X\to \mathbb {C} } ) be continuous.[4]

## Compact support

Functions with ***compact support*** on a topological space X {\displaystyle X} are those whose closed support is a [compact](/source/Compact_space) subset of X . {\displaystyle X.} If X {\displaystyle X} is the real line, or n {\displaystyle n} -dimensional Euclidean space, then a function has compact support if and only if it has ***bounded support***, since a subset of R n {\displaystyle \mathbb {R} ^{n}} is compact if and only if it is closed and bounded.

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

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

Real-valued compactly supported [smooth functions](/source/Smooth_function) on a [Euclidean space](/source/Euclidean_space) are called [bump functions](/source/Bump_function). [Mollifiers](/source/Mollifier) are an important special case of bump functions as they can be used in [distribution theory](/source/Distribution_(mathematics)) to create [sequences](/source/Sequence) of smooth functions approximating nonsmooth (generalized) functions, via [convolution](/source/Convolution).

In [good cases](/source/Well-behaved), functions with compact support are [dense](/source/Dense_set) 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 [limits](/source/Limit_(mathematics)), for any ε > 0 , {\displaystyle \varepsilon >0,} any function f {\displaystyle f} on the real line R {\displaystyle \mathbb {R} } that vanishes at infinity can be approximated by choosing an appropriate compact subset C {\displaystyle C} of R {\displaystyle \mathbb {R} } such that | f ( x ) − I C ( x ) f ( x ) | < ε {\displaystyle \left|f(x)-I_{C}(x)f(x)\right|<\varepsilon } for all x ∈ X , {\displaystyle x\in X,} where I C {\displaystyle I_{C}} is the [indicator function](/source/Indicator_function) of C . {\displaystyle 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 {\displaystyle X} is a topological [measure space](/source/Measure_space) with a [Borel measure](/source/Borel_measure) μ {\displaystyle \mu } (such as R n , {\displaystyle \mathbb {R} ^{n},} or a [Lebesgue measurable](/source/Lebesgue_measure) subset of R n , {\displaystyle \mathbb {R} ^{n},} equipped with Lebesgue measure), then one typically identifies functions that are equal μ {\displaystyle \mu } -almost everywhere. In that case, the ***essential support*** of a measurable function f : X → R {\displaystyle f:X\to \mathbb {R} } written **e s s s u p p ⁡ ( f ) , {\displaystyle \operatorname {ess\,supp} (f),}** is defined to be the smallest closed subset F {\displaystyle F} of X {\displaystyle X} such that f = 0 {\displaystyle f=0} μ {\displaystyle \mu } -almost everywhere outside F . {\displaystyle F.} Equivalently, e s s s u p p ⁡ ( f ) {\displaystyle \operatorname {ess\,supp} (f)} is the complement of the largest [open set](/source/Open_set) on which f = 0 {\displaystyle f=0} μ {\displaystyle \mu } -almost everywhere[5] e s s s u p p ⁡ ( f ) := X ∖ ⋃ { Ω ⊆ X : Ω is open and f = 0 μ -almost everywhere in Ω } . {\displaystyle \operatorname {ess\,supp} (f):=X\setminus \bigcup \left\{\Omega \subseteq X:\Omega {\text{ is open and }}f=0\,\mu {\text{-almost everywhere in }}\Omega \right\}.}

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

## Generalization

If M {\displaystyle M} is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X → M . {\displaystyle f:X\to M.} Support may also be defined for any [algebraic structure](/source/Algebraic_structure) with [identity](/source/Identity_element) (such as a [group](/source/Group_(mathematics)), [monoid](/source/Monoid), or [composition algebra](/source/Composition_algebra)), in which the identity element assumes the role of zero. For instance, the family Z N {\displaystyle \mathbb {Z} ^{\mathbb {N} }} of functions from the [natural numbers](/source/Natural_numbers) to the [integers](/source/Integers) is the [uncountable](/source/Uncountable) set of integer sequences. The subfamily { f ∈ Z N : f has finite support } {\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 as [group rings](/source/Group_ring) and [free abelian groups](/source/Free_abelian_group).[7]

## In probability and measure theory

Further information: [Support (measure theory)](/source/Support_(measure_theory))

In [probability theory](/source/Probability_theory), the support of a [probability distribution](/source/Probability_distribution) can be loosely thought of as the [closure](/source/Closure_(mathematics)) 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](/source/Sigma_algebra), rather than on a topological space.

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

In practice however, the support of a [discrete random variable](/source/Discrete_random_variable) X {\displaystyle X} is often defined as the set R X = { x ∈ R : P ( X = x ) > 0 } {\displaystyle R_{X}=\{x\in \mathbb {R} :P(X=x)>0\}} and the support of a [continuous random variable](/source/Continuous_random_variable) X {\displaystyle X} is defined as the set R X = { x ∈ R : f X ( x ) > 0 } {\displaystyle R_{X}=\{x\in \mathbb {R} :f_{X}(x)>0\}} where f X ( x ) {\displaystyle f_{X}(x)} is a [probability density function](/source/Probability_density_function) of X {\displaystyle X} (the [set-theoretic support](#set-theoretic_support)).[8]

Note that the word *support* can refer to the [logarithm](/source/Logarithm) of the [likelihood](/source/Likelihood_function) of a probability density function.[9]

## Support of a distribution

It is possible also to talk about the support of a [distribution](/source/Distribution_(mathematics)), such as the [Dirac delta function](/source/Dirac_delta_function) δ ( x ) {\displaystyle \delta (x)} on the real line. In that example, we can consider test functions F , {\displaystyle F,} which are [smooth functions](/source/Smooth_function) with support not including the point 0. {\displaystyle 0.} Since δ ( F ) {\displaystyle \delta (F)} (the distribution δ {\displaystyle \delta } applied as [linear functional](/source/Linear_functional) to F {\displaystyle F} ) is 0 {\displaystyle 0} for such functions, we can say that the support of δ {\displaystyle \delta } is { 0 } {\displaystyle \{0\}} only. Since measures (including [probability measures](/source/Probability_measure)) 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 {\displaystyle f} is a distribution, and that U {\displaystyle U} is an open set in Euclidean space such that, for all test functions ϕ {\displaystyle \phi } such that the support of ϕ {\displaystyle \phi } is contained in U , {\displaystyle U,} f ( ϕ ) = 0. {\displaystyle f(\phi )=0.} Then f {\displaystyle f} is said to vanish on U . {\displaystyle U.} Now, if f {\displaystyle f} vanishes on an arbitrary family U α {\displaystyle U_{\alpha }} of open sets, then for any test function ϕ {\displaystyle \phi } supported in ⋃ U α , {\textstyle \bigcup U_{\alpha },} a simple argument based on the compactness of the support of ϕ {\displaystyle \phi } and a partition of unity shows that f ( ϕ ) = 0 {\displaystyle f(\phi )=0} as well. Hence we can define the *support* of f {\displaystyle f} as the complement of the largest open set on which f {\displaystyle f} vanishes. For example, the support of the Dirac delta is { 0 } . {\displaystyle \{0\}.}

## Singular support

In [Fourier analysis](/source/Fourier_analysis) in particular, it is interesting to study the ***singular support*** of a distribution. This has the intuitive interpretation as the set of points at which a distribution *fails to be a smooth function*.

For example, the [Fourier transform](/source/Fourier_transform) of the [Heaviside step function](/source/Heaviside_step_function) can, up to constant factors, be considered to be 1 / x {\displaystyle 1/x} (a function) *except* at x = 0. {\displaystyle x=0.} While x = 0 {\displaystyle x=0} is clearly a special point, it is more precise to say that the transform of the distribution has singular support { 0 } {\displaystyle \{0\}} : it cannot accurately be expressed as a function in relation to test functions with support including 0. {\displaystyle 0.} It *can* be expressed as an application of a [Cauchy principal value](/source/Cauchy_principal_value) *improper* integral.

For distributions in several variables, singular supports allow one to define *[wave front sets](/source/Wave_front_set)* and understand [Huygens' principle](/source/Huygens'_principle) in terms of [mathematical analysis](/source/Mathematical_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).

## Family of supports

An abstract notion of ***family of supports*** on a [topological space](/source/Topological_space) X , {\displaystyle X,} suitable for [sheaf theory](/source/Sheaf_theory), was defined by [Henri Cartan](/source/Henri_Cartan). In extending [Poincaré duality](/source/Poincar%C3%A9_duality) to [manifolds](/source/Manifold) that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example [Alexander–Spanier cohomology](/source/Alexander%E2%80%93Spanier_cohomology).

Bredon, *Sheaf Theory* (2nd edition, 1997) gives these definitions. A family Φ {\displaystyle \Phi } of closed subsets of X {\displaystyle X} is a *family of supports*, if it is [down-closed](https://en.wikipedia.org/w/index.php?title=Down-closed&action=edit&redlink=1) and closed under [finite union](/source/Finite_union). Its *extent* is the union over Φ . {\displaystyle \Phi .} A *paracompactifying* family of supports that satisfies further that any Y {\displaystyle Y} in Φ {\displaystyle \Phi } is, with the [subspace topology](/source/Subspace_topology), a [paracompact space](/source/Paracompact_space); and has some Z {\displaystyle Z} in Φ {\displaystyle \Phi } which is a [neighbourhood](/source/Neighbourhood_(topology)). If X {\displaystyle X} is a [locally compact space](/source/Locally_compact_space), assumed [Hausdorff](/source/Hausdorff_space), the family of all [compact subsets](/source/Compact_subset) satisfies the further conditions, making it paracompactifying.

## See also

- [Bounded function](/source/Bounded_function) – Mathematical function whose set of values is bounded

- [Bump function](/source/Bump_function) – Smooth and compactly supported function

- [Support of a module](/source/Support_of_a_module)

- [Titchmarsh convolution theorem](/source/Titchmarsh_convolution_theorem)

## Citations

1. **[^](#cite_ref-folland_1-0)** Folland, Gerald B. (1999). *Real Analysis, 2nd ed*. New York: John Wiley. p. 132.

1. **[^](#cite_ref-hormander_2-0)** [Hörmander, Lars](/source/Lars_H%C3%B6rmander) (1990). *Linear Partial Differential Equations I, 2nd ed*. Berlin: Springer-Verlag. p. 14.

1. **[^](#cite_ref-Pasc_3-0)** Pascucci, Andrea (2011). *PDE and Martingale Methods in Option Pricing*. Bocconi & Springer Series. Berlin: Springer-Verlag. p. 678. [doi](/source/Doi_(identifier)):[10.1007/978-88-470-1781-8](https://doi.org/10.1007%2F978-88-470-1781-8). [ISBN](/source/ISBN_(identifier)) [978-88-470-1780-1](https://en.wikipedia.org/wiki/Special:BookSources/978-88-470-1780-1).

1. **[^](#cite_ref-4)** Rudin, Walter (1987). *Real and Complex Analysis, 3rd ed*. New York: McGraw-Hill. p. 38.

1. ^ [***a***](#cite_ref-lieb_5-0) [***b***](#cite_ref-lieb_5-1) [Lieb, Elliott](/source/Elliott_H._Lieb); [Loss, Michael](/source/Michael_Loss) (2001). *Analysis*. Graduate Studies in Mathematics. Vol. 14 (2nd ed.). [American Mathematical Society](/source/American_Mathematical_Society). p. 13. [ISBN](/source/ISBN_(identifier)) [978-0821827833](https://en.wikipedia.org/wiki/Special:BookSources/978-0821827833).

1. **[^](#cite_ref-6)** In a similar way, one uses the [essential supremum](/source/Essential_supremum_and_essential_infimum) of a measurable function instead of its supremum.

1. **[^](#cite_ref-7)** Tomasz, Kaczynski (2004). *Computational homology*. Mischaikow, Konstantin Michael,, Mrozek, Marian. New York: Springer. p. 445. [ISBN](/source/ISBN_(identifier)) [9780387215976](https://en.wikipedia.org/wiki/Special:BookSources/9780387215976). [OCLC](/source/OCLC_(identifier)) [55897585](https://search.worldcat.org/oclc/55897585).

1. **[^](#cite_ref-8)** Taboga, Marco. ["Support of a random variable"](https://www.statlect.com/glossary/support-of-a-random-variable). *statlect.com*. Retrieved 29 November 2017.

1. **[^](#cite_ref-9)** Edwards, A. W. F. (1992). [*Likelihood*](https://books.google.com/books?id=LL08AAAAIAAJ&pg=PA31) (Expanded ed.). Baltimore: Johns Hopkins University Press. pp. 31–34. [ISBN](/source/ISBN_(identifier)) [0-8018-4443-6](https://en.wikipedia.org/wiki/Special:BookSources/0-8018-4443-6).

## References

- [Rudin, Walter](/source/Walter_Rudin) (1991). [*Functional Analysis*](https://archive.org/details/functionalanalys00rudi). International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: [McGraw-Hill Science/Engineering/Math](/source/McGraw-Hill_Science%2FEngineering%2FMath). [ISBN](/source/ISBN_(identifier)) [978-0-07-054236-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-054236-5). [OCLC](/source/OCLC_(identifier)) [21163277](https://search.worldcat.org/oclc/21163277).

- [Trèves, François](/source/Fran%C3%A7ois_Tr%C3%A8ves) (2006) [1967]. *Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. [ISBN](/source/ISBN_(identifier)) [978-0-486-45352-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-45352-1). [OCLC](/source/OCLC_(identifier)) [853623322](https://search.worldcat.org/oclc/853623322).

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