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Formal definition

Let f : E → F {\displaystyle f:E\to F} be a function from a set E {\displaystyle E} to a set F . {\displaystyle F.} If a set A {\displaystyle A} is a subset of E , {\displaystyle E,} then the restriction of f {\displaystyle f} to A {\displaystyle A} is the function¹ f | A : A → F {\displaystyle {f|}_{A}:A\to F} given by f | A ( x ) = f ( x ) {\displaystyle {f|}_{A}(x)=f(x)} for x ∈ A . {\displaystyle x\in A.} Informally, the restriction of f {\displaystyle f} to A {\displaystyle A} is the same function as f , {\displaystyle f,} but is only defined on A {\displaystyle A} .

If the function f {\displaystyle f} is thought of as a relation ( x , f ( x ) ) {\displaystyle (x,f(x))} on the Cartesian product E × F , {\displaystyle E\times F,} then the restriction of f {\displaystyle f} to A {\displaystyle A} can be represented by its graph,

G ( f | A ) = { ( x , f ( x ) ) ∈ G ( f ) : x ∈ A } = G ( f ) ∩ ( A × F ) , {\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f):x\in A\}=G(f)\cap (A\times F),}

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

Extensions

A function F {\displaystyle F} is said to be an extension of another function f {\displaystyle f} if whenever x {\displaystyle x} is in the domain of f {\displaystyle f} then x {\displaystyle x} is also in the domain of F {\displaystyle F} and f ( x ) = F ( x ) . {\displaystyle f(x)=F(x).} That is, if domain ⁡ f ⊆ domain ⁡ F {\displaystyle \operatorname {domain} f\subseteq \operatorname {domain} F} and F | domain ⁡ f = f . {\displaystyle F{\big \vert }_{\operatorname {domain} f}=f.}

A linear extension (respectively, continuous extension, etc.) of a function f {\displaystyle f} is an extension of f {\displaystyle f} that is also a linear map (respectively, a continuous map, etc.).

Examples

1. The restriction of the non-injective function f : R → R ,   x ↦ x 2 {\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto x^{2}} to the domain R + = [ 0 , ∞ ) {\displaystyle \mathbb {R} _{+}=[0,\infty )} is the injection f : R + → R ,   x ↦ x 2 . {\displaystyle f:\mathbb {R} _{+}\to \mathbb {R} ,\ x\mapsto x^{2}.}
2. The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: Γ | Z + ( n ) = ( n − 1 ) ! {\displaystyle {\Gamma |}_{\mathbb {Z} ^{+}}\!(n)=(n-1)!}

Properties of restrictions

• Restricting a function f : X → Y {\displaystyle f:X\rightarrow Y} to its entire domain X {\displaystyle X} gives back the original function, that is, f | X = f . {\displaystyle f|_{X}=f.}
• Restricting a function twice is the same as restricting it once, that is, if A ⊆ B ⊆ dom ⁡ f , {\displaystyle A\subseteq B\subseteq \operatorname {dom} f,} then ( f | B ) | A = f | A . {\displaystyle \left(f|_{B}\right)|_{A}=f|_{A}.}
• The restriction of the identity function on a set X {\displaystyle X} to a subset A {\displaystyle A} of X {\displaystyle X} is just the inclusion map from A {\displaystyle A} into X . {\displaystyle X.} ²
• The restriction of a continuous function is continuous.³⁴

Applications

Inverse functions

Main article: Inverse function

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

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

Selection operators

Main article: Selection (relational algebra)

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as σ a θ b ( R ) {\displaystyle \sigma _{a\theta b}(R)} or σ a θ v ( R ) {\displaystyle \sigma _{a\theta v}(R)} where:

• a {\displaystyle a} and b {\displaystyle b} are attribute names,
• θ {\displaystyle \theta } is a binary operation in the set { < , ≤ , = , ≠ , ≥ , > } , {\displaystyle \{<,\leq ,=,\neq ,\geq ,>\},}
• v {\displaystyle v} is a value constant,
• R {\displaystyle R} is a relation.

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

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

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

The pasting lemma

Main article: Pasting lemma

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

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

Main article: Sheaf theory

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

In sheaf theory, one assigns an object F ( U ) {\displaystyle F(U)} in a category to each open set U {\displaystyle U} of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested open sets; that is, if V ⊆ U , {\displaystyle V\subseteq U,} then there is a morphism res V , U : F ( U ) → F ( V ) {\displaystyle \operatorname {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 U {\displaystyle U} of X , {\displaystyle X,} the restriction morphism res U , U : F ( U ) → F ( U ) {\displaystyle \operatorname {res} _{U,U}:F(U)\to F(U)} is the identity morphism on F ( U ) . {\displaystyle F(U).}
• If we have three open sets W ⊆ V ⊆ U , {\displaystyle W\subseteq V\subseteq U,} then the composite res W , V ∘ res V , U = res W , U . {\displaystyle \operatorname {res} _{W,V}\circ \operatorname {res} _{V,U}=\operatorname {res} _{W,U}.}
• (Locality) If ( U i ) {\displaystyle \left(U_{i}\right)} is an open covering of an open set U , {\displaystyle U,} and if s , t ∈ F ( U ) {\displaystyle s,t\in F(U)} are such that s | U i = t | U i {\displaystyle s{\big \vert }_{U_{i}}=t{\big \vert }_{U_{i}}} for each set U i {\displaystyle U_{i}} of the covering, then s = t {\displaystyle s=t} ; and
• (Gluing) If ( U i ) {\displaystyle \left(U_{i}\right)} is an open covering of an open set U , {\displaystyle U,} and if for each i {\displaystyle i} a section x i ∈ F ( U i ) {\displaystyle x_{i}\in F\left(U_{i}\right)} is given such that for each pair U i , U j {\displaystyle U_{i},U_{j}} of the covering sets the restrictions of s i {\displaystyle s_{i}} and s j {\displaystyle s_{j}} agree on the overlaps: s i | U i ∩ U j = s j | U i ∩ U j , {\displaystyle s_{i}{\big \vert }_{U_{i}\cap U_{j}}=s_{j}{\big \vert }_{U_{i}\cap U_{j}},} then there is a section s ∈ F ( U ) {\displaystyle s\in F(U)} such that s | U i = s i {\displaystyle s{\big \vert }_{U_{i}}=s_{i}} for each i . {\displaystyle i.}

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

Left- and right-restriction

More generally, the restriction (or domain restriction or left-restriction) A ◃ R {\displaystyle A\triangleleft R} of a binary relation R {\displaystyle R} between E {\displaystyle E} and F {\displaystyle F} may be defined as a relation having domain A , {\displaystyle A,} codomain F {\displaystyle F} and graph G ( A ◃ R ) = { ( x , y ) ∈ F ( R ) : x ∈ A } . {\displaystyle G(A\triangleleft R)=\{(x,y)\in F(R):x\in A\}.} Similarly, one can define a right-restriction or range restriction R ▹ B . {\displaystyle R\triangleright B.} Indeed, one could define a restriction to n {\displaystyle n} -ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product E × F {\displaystyle E\times F} for binary relations. These cases do not fit into the scheme of sheaves.

Anti-restriction

The domain anti-restriction (or domain subtraction) of a function or binary relation R {\displaystyle R} (with domain E {\displaystyle E} and codomain F {\displaystyle F} ) by a set A {\displaystyle A} may be defined as ( E ∖ A ) ◃ R {\displaystyle (E\setminus A)\triangleleft R} ; it removes all elements of A {\displaystyle A} from the domain E . {\displaystyle E.} It is sometimes denoted A {\displaystyle A}  ⩤  R . {\displaystyle R.} ⁵ Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R {\displaystyle R} by a set B {\displaystyle B} is defined as R ▹ ( F ∖ B ) {\displaystyle R\triangleright (F\setminus B)} ; it removes all elements of B {\displaystyle B} from the codomain F . {\displaystyle F.} It is sometimes denoted R {\displaystyle R}  ⩥  B . {\displaystyle B.}

See also

• Constraint – Condition of an optimization problem which the solution must satisfy
• Deformation retract – Continuous, position-preserving mapping from a topological space into a subspacePages displaying short descriptions of redirect targets
• Local property – property which occurs on sufficiently small or arbitrarily small neighborhoods of pointsPages displaying wikidata descriptions as a fallback
• Function (mathematics) § Restriction and extension
• Binary relation § Restriction
• Relational algebra § Selection (σ)

References

1. Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. [36]. ISBN 0-7167-0457-9. 0-7167-0457-9 ↩

2. Halmos, Paul (1960). Naive Set Theory. Princeton, NJ: D. Van Nostrand. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition). /wiki/Paul_Halmos ↩

3. Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2. 0-13-181629-2 ↩

4. Adams, Colin Conrad; Franzosa, Robert David (2008). Introduction to Topology: Pure and Applied. Pearson Prentice Hall. ISBN 978-0-13-184869-6. 978-0-13-184869-6 ↩

5. Dunne, S. and Stoddart, Bill Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 5–7, 2006, Revised Selected ... Computer Science and General Issues). Springer (2006) ↩