Restriction (mathematics)
Function  

x ↦ f (x)  
By domain and codomain  


Classes/properties  
Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective  
Constructions  
Restriction · Composition · λ · Inverse  
Generalizations  
Partial · Multivalued · Implicit  
In mathematics, the restriction of a function f is a new function f_{A} obtained by choosing a smaller domain A for the original function f. The notation is also used.
Formal definition
Let f : E → F be a function from a set E to a set F, so that the domain of f is in E (). If a set A is a subset of E, then the restriction of f to A is the function^{[1]}
 .
Informally, the restriction of f to A is the same function as f, but is only defined on .
If the function f is thought of as a relation on the Cartesian product , then the restriction of f to A can be represented by the graph , where the pairs represent edges in the graph G.
Examples
 The restriction of the noninjective function to is the injection .
 The factorial function is the restriction of the gamma function to the integers.
Properties of restrictions
 Restricting a function to its entire domain gives back the original function; i.e., .
 Restricting a function twice is the same as restricting it once; i.e. if , then .
 The restriction of the identity function on a space X to a subset A of X is just the inclusion map of A into X.^{[2]}
 The restriction of a continuous function is continuous.^{[3]}^{[4]}
Applications
Inverse functions
For a function to have an inverse, it must be onetoone. If a function f is not onetoone, it may be possible to define a partial inverse of f by restricting the domain. For example, the function
is not onetoone, since x^{2} = (−x)^{2}. However, the function becomes onetoone if we restrict to the domain x ≥ 0, in which case
(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:
Selection operators
In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as or where:
 and are attribute names
 is a binary operation in the set
 is a value constant
 is a relation
The selection selects all those tuples in for which holds between the and the attribute.
The selection selects all those tuples in for which holds between the attribute and the value .
Thus, the selection operator restricts to a subset of the entire database.
The 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 be both closed (or both open) subsets of a topological space A such that , and let B also be a topological space. If is continuous when restricted to both X and Y, then 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
Sheaves provide a way of generalizing restrictions to objects besides functions.
In sheaf theory, one assigns an object in a category to each open set 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; i.e., if , then there is a morphism res_{V,U} : F(U) → F(V) satisfying the following properties, which are designed to mimic the restriction of a function:
 For every open set U of X, the restriction morphism res_{U,U} : F(U) → F(U) is the identity morphism on F(U).
 If we have three open sets W ⊆ V ⊆ U, then the composite res_{W,V} o res_{V,U} = res_{W,U}.
 (Locality) If (U_{i}) is an open covering of an open set U, and if s,t ∈ F(U) are such that s_{Ui} = t_{Ui} for each set U_{i} of the covering, then s = t; and
 (Gluing) If (U_{i}) is an open covering of an open set U, and if for each i a section s_{i} ∈ F(U_{i}) is given such that for each pair U_{i},U_{j} of the covering sets the restrictions of s_{i} and s_{j} agree on the overlaps: s_{i}_{Ui∩Uj} = s_{j}_{Ui∩Uj}, then there is a section s ∈ F(U) such that s_{Ui} = s_{i} for each i.
The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a presheaf.
Left and rightrestriction
More generally, the restriction (or domain restriction or leftrestriction) A ◁ R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G(A ◁ R) = {(x, y) ∈ G(R)  x ∈ A} . Similarly, one can define a rightrestriction or range restriction R ▷ B. Indeed, one could define a restriction to nary relations, as well as to subsets understood as relations, such as ones of E × F for binary relations. These cases do not fit into the scheme of sheaves.
Antirestriction
The domain antirestriction (or domain subtraction) of a function or binary relation R (with domain E and codomain F) by a set A may be defined as (E \ A) ◁ R; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R.^{[5]} Similarly, the range antirestriction (or range subtraction) of a function or binary relation R by a set B is defined as R ▷ (F \ B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.
See also
 Deformation retract
 Function (mathematics) #Restrictions and extensions
 Binary relation #Restriction
 Relational algebra#Selection (σ)
References
 ↑ Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.
 ↑ Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by SpringerVerlag, New York, 1974. ISBN 0387900926 (SpringerVerlag edition). Reprinted by Martino Fine Books, 2011. ISBN 9781614271314 (Paperback edition).
 ↑ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
 ↑ Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
 ↑ Dunne, S. and Stoddart, Bill Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 57, 2006, Revised Selected ... Computer Science and General Issues). Springer (2006)