In mathematics, the restriction of a function f is a new function, denoted f\vert_A or obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert_A.

Formal definition

Let f : E **to **F be a function from a set E **to **a set F. If a set A is a subset of E, then the **restriction of **f **to **A is the function given by for x \in A. Informally, the restriction of f to A is the same function as f, but is only defined on A. If the function f is thought of as a relation (x,f(x)) on the Cartesian product E \times F, then the restriction of f to A can be represented by its graph, where the pairs (x,f(x)) represent ordered pairs in the graph G.

Extensions

A function F is said to be an of another function f if whenever x is in the domain of f then x is also in the domain of F and That is, if and A Linear extension of a function (respectively, Continuous extension, etc.) of a function f is an extension of f that is also a linear map (respectively, a continuous map, etc.).

Examples

Properties of restrictions

Applications

Inverse functions

For a function to have an inverse, it must be one-to-one. If a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function f(x) = x^2 defined on the whole of \R is not one-to-one since for any x \in \R. However, the function becomes one-to-one if we restrict to the domain in which case (If we instead restrict to the domain then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we allow the inverse to be 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: The selection selects all those tuples in R for which \theta holds between the a and the b attribute. The selection selects all those tuples in R for which \theta holds between the a attribute and the value v. 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 X,Y be two closed subsets (or two open subsets) of a topological space A such that and let B also be a topological space. If f: A \to B 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 F(U) in a category to each open set 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 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.

Left- and right-restriction

More generally, the restriction (or domain restriction or left-restriction) of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph Similarly, one can define a right-restriction or range restriction Indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product 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 (with domain E and codomain F) by a set A may be defined as ; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R. Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as ; it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.