|x ↦ f (x)|
|Examples of domains and codomains|
In mathematics, the restriction of a function is a new function, denoted or obtained by choosing a smaller domain for the original function The function is then said to extend
Let be a function from a set to a set If a set is a subset of then the restriction of to is the function
If the function is thought of as a relation on the Cartesian product then the restriction of to can be represented by its graph where the pairs represent ordered pairs in the graph
A function is said to be an extension of another function if whenever is in the domain of then is also in the domain of and That is, if and
A linear extension (respectively, continuous extension, etc.) of a function is an extension of that is also a linear map (respectively, a continuous map, etc.).
Main article: Inverse function
For a function to have an inverse, it must be one-to-one. If a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function
(If we instead restrict to the domain then the inverse is the negative of the square root of ) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.
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 or where:
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.
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 be two closed subsets (or two open subsets) of a topological space such that and let also be a topological space. If is continuous when restricted to both and then 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.
Main article: Sheaf theory
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; that is, if then there is a morphism satisfying the following properties, which are designed to mimic the restriction of a function:
The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf.
More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain codomain and graph Similarly, one can define a right-restriction or range restriction Indeed, one could define a restriction to -ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product for binary relations. These cases do not fit into the scheme of sheaves.[clarification needed]
The domain anti-restriction (or domain subtraction) of a function or binary relation (with domain and codomain ) by a set may be defined as ; it removes all elements of from the domain It is sometimes denoted ⩤  Similarly, the range anti-restriction (or range subtraction) of a function or binary relation by a set is defined as ; it removes all elements of from the codomain It is sometimes denoted ⩥