Inmathematics, therestriction of afunction${\displaystyle f}$ is a new function, denoted${\displaystyle f{|}_A}$ or${\displaystyle f{\upharpoonright }_A,}$ obtained by choosing a smallerdomain${\displaystyle A}$ for the original function${\displaystyle f.}$ The function${\displaystyle f}$ is then said toextend${\displaystyle f{|}_A.}$

Let${\displaystyle f:E\to F}$ be a function from aset${\displaystyle E}$ to a set${\displaystyle F.}$ If a set${\displaystyle A}$ is asubset of${\displaystyle E,}$ then therestriction of${\displaystyle f}$to${\displaystyle A}$ is the function^[1]$${\displaystyle {f|}_A:A\to F}$$given by${\displaystyle {f|}_A(x)=f(x)}$ for${\displaystyle x\in A.}$ Informally, the restriction of${\displaystyle f}$ to${\displaystyle A}$ is the same function as${\displaystyle f,}$ but is only defined on${\displaystyle A}$.

If the function${\displaystyle f}$ is thought of as arelation${\displaystyle (x,f(x))}$ on theCartesian product${\displaystyle E\times F,}$ then the restriction of${\displaystyle f}$ to${\displaystyle A}$ can be represented by itsgraph,

${\displaystyle G({f|}_A)=\{(x,f(x))\in G(f):x\in A\}=G(f)\cap (A\times F),}$

where the pairs${\displaystyle (x,f(x))}$ representordered pairs in the graph${\displaystyle G.}$

A function${\displaystyle F}$ is said to be anextension of another function${\displaystyle f}$ if whenever${\displaystyle x}$ is in the domain of${\displaystyle f}$ then${\displaystyle x}$ is also in the domain of${\displaystyle F}$ and${\displaystyle f(x)=F(x).}$ That is, if${\displaystyle \mathrm{domain}f\subseteq \mathrm{domain}F}$ and${\displaystyle F{|}_{\mathrm{domain}f}=f.}$

Alinear extension (respectively,continuous extension, etc.) of a function${\displaystyle f}$ is an extension of${\displaystyle f}$ that is also alinear map (respectively, acontinuous map, etc.).

1. The restriction of thenon-injective function${\displaystyle f:\mathbb{R}\to \mathbb{R},x\mapsto x^2}$ to the domain${\displaystyle {\mathbb{R}}_{+}=[0,\mathrm{\infty })}$ is the injection${\displaystyle f:{\mathbb{R}}_{+}\to \mathbb{R},x\mapsto x^2.}$
2. Thefactorial function is the restriction of thegamma function to the positive integers, with the argument shifted by one:${\displaystyle {\mathrm{\Gamma }|}_{{\mathbb{Z}}^{+}}(n)=(n-1)!}$

• Restricting a function${\displaystyle f:X\to Y}$ to its entire domain${\displaystyle X}$ gives back the original function, that is,${\displaystyle f{|}_X=f.}$
• Restricting a function twice is the same as restricting it once, that is, if${\displaystyle A\subseteq B\subseteq \mathrm{dom}f,}$ then${\displaystyle \left(f{|}_B\right){|}_A=f{|}_A.}$
• The restriction of theidentity function on a set${\displaystyle X}$ to a subset${\displaystyle A}$ of${\displaystyle X}$ is just theinclusion map from${\displaystyle A}$ into${\displaystyle X.}$^[2]
• The restriction of acontinuous function is continuous.^[3][4]

For a function to have an inverse, it must beone-to-one. If a function${\displaystyle f}$ is not one-to-one, it may be possible to define apartial inverse of${\displaystyle f}$ by restricting the domain. For example, the function$${\displaystyle f(x)=x^2}$$defined on the whole of${\displaystyle \mathbb{R}}$ is not one-to-one since${\displaystyle x^2=(-x{)}^2}$ for any${\displaystyle x\in \mathbb{R}.}$ However, the function becomes one-to-one if we restrict to the domain${\displaystyle {\mathbb{R}}_{\ge 0}=[0,\mathrm{\infty }),}$ in which case$${\displaystyle f^{-1}(y)=\sqrt{y}.}$$

(If we instead restrict to the domain${\displaystyle (-\mathrm{\infty },0],}$ then the inverse is the negative of the square root of${\displaystyle y.}$) Alternatively, there is no need to restrict the domain if we allow the inverse to be amultivalued function.

Inrelational algebra, aselection (sometimes called a restriction to avoid confusion withSQL's use of SELECT) is aunary operation written as${\displaystyle {\sigma }_{a\theta b}(R)}$ or${\displaystyle {\sigma }_{a\theta v}(R)}$ where:

• ${\displaystyle a}$ and${\displaystyle b}$ are attribute names,
• ${\displaystyle \theta }$ is abinary operation in the set
• ${\displaystyle v}$ is a value constant,
• ${\displaystyle R}$ is arelation.

The selection${\displaystyle {\sigma }_{a\theta b}(R)}$ selects all thosetuples in${\displaystyle R}$ for which${\displaystyle \theta }$ holds between the${\displaystyle a}$ and the${\displaystyle b}$ attribute.

The selection${\displaystyle {\sigma }_{a\theta v}(R)}$ selects all those tuples in${\displaystyle R}$ for which${\displaystyle \theta }$ holds between the${\displaystyle a}$ attribute and the value${\displaystyle v.}$

Thus, the selection operator restricts to a subset of the entire database.

The pasting lemma is a result intopology that relates the continuity of a function with the continuity of its restrictions to subsets.

Let${\displaystyle X,Y}$ be two closed subsets (or two open subsets) of a topological space${\displaystyle A}$ such that${\displaystyle A=X\cup Y,}$ and let${\displaystyle B}$ also be a topological space. If${\displaystyle f:A\to B}$ is continuous when restricted to both${\displaystyle X}$ and${\displaystyle Y,}$ then${\displaystyle f}$ is continuous.

This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.

Sheaves provide a way of generalizing restrictions to objects besides functions.

Insheaf theory, one assigns an object${\displaystyle F(U)}$ in acategory to eachopen set${\displaystyle U}$ of atopological space, and requires that the objects satisfy certain conditions. The most important condition is that there arerestrictionmorphisms between every pair of objects associated to nested open sets; that is, if${\displaystyle V\subseteq U,}$ then there is a morphism${\displaystyle {\mathrm{res}}_{V,U}:F(U)\to F(V)}$ satisfying the following properties, which are designed to mimic the restriction of a function:

• For every open set${\displaystyle U}$ of${\displaystyle X,}$ the restriction morphism${\displaystyle {\mathrm{res}}_{U,U}:F(U)\to F(U)}$ is the identity morphism on${\displaystyle F(U).}$
• If we have three open sets${\displaystyle W\subseteq V\subseteq U,}$ then thecomposite${\displaystyle {\mathrm{res}}_{W,V}\circ {\mathrm{res}}_{V,U}={\mathrm{res}}_{W,U}.}$
• (Locality) If${\displaystyle \left(U_i\right)}$ is an opencovering of an open set${\displaystyle U,}$ and if${\displaystyle s,t\in F(U)}$ are such that${\displaystyle s{|}_{U_i}=t{|}_{U_i}}$ for each set${\displaystyle U_i}$ of the covering, then${\displaystyle s=t}$; and
• (Gluing) If${\displaystyle \left(U_i\right)}$ is an open covering of an open set${\displaystyle U,}$ and if for each${\displaystyle i}$ a section${\displaystyle x_i\in F\left(U_i\right)}$ is given such that for each pair${\displaystyle U_i,U_j}$ of the covering sets the restrictions of${\displaystyle s_i}$ and${\displaystyle s_j}$ agree on the overlaps:${\displaystyle s_i{|}_{U_i\cap U_j}=s_j{|}_{U_i\cap U_j},}$ then there is a section${\displaystyle s\in F(U)}$ such that${\displaystyle s{|}_{U_i}=s_i}$ for each${\displaystyle i.}$

The collection of all such objects is called asheaf. If only the first two properties are satisfied, it is apre-sheaf.

More generally, the restriction (ordomain restriction orleft-restriction)${\displaystyle A◃R}$ of abinary relation${\displaystyle R}$ between${\displaystyle E}$ and${\displaystyle F}$ may be defined as a relation having domain${\displaystyle A,}$ codomain${\displaystyle F}$ and graph${\displaystyle G(A◃R)=\{(x,y)\in F(R):x\in A\}.}$ Similarly, one can define aright-restriction orrange restriction${\displaystyle R▹B.}$ Indeed, one could define a restriction to${\displaystyle n}$-ary relations, as well as tosubsets understood as relations, such as ones of theCartesian product${\displaystyle E\times F}$ for binary relations.These cases do not fit into the scheme ofsheaves.^{[clarification needed]}

Thedomain anti-restriction (ordomain subtraction) of a function or binary relation${\displaystyle R}$ (with domain${\displaystyle E}$ and codomain${\displaystyle F}$) by a set${\displaystyle A}$ may be defined as${\displaystyle (E\setminus A)◃R}$; it removes all elements of${\displaystyle A}$ from the domain${\displaystyle E.}$ It is sometimes denoted${\displaystyle A}$ ⩤ ${\displaystyle R.}$^[5] Similarly, therange anti-restriction (orrange subtraction) of a function or binary relation${\displaystyle R}$ by a set${\displaystyle B}$ is defined as${\displaystyle R▹(F\setminus B)}$; it removes all elements of${\displaystyle B}$ from the codomain${\displaystyle F.}$ It is sometimes denoted${\displaystyle R}$ ⩥ ${\displaystyle B.}$

• Constraint – Condition of an optimization problem which the solution must satisfy
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• Relational algebra § Selection (σ)

• Sheaf theory

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