In mathematics, the restriction of a function obtained by choosing a smaller domain is then said to extend

Formal definition

be a function from a set is a subset then the restriction of to is the function.

is thought of as a relation on the Cartesian product can be represented by its graph,

represent ordered pairs

Extensions

is said to be an

A linear extension (respectively, continuous extension that is also a linear map (respectively, a continuous map, etc.).

Examples

1. The restriction of the non-injective

2. The factorial function is the restriction of the gamma function

Properties of restrictions

• The restriction of the identity function is just the inclusion map

• The restriction of a continuous function is continuous.

Applications

Inverse functions

For a function to have an inverse, it must be one-to-one is not one-to-one, it may be possible to define a partial inverse

) 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 where:

• are attribute names,

• is a binary operation

• is a value constant,

• is a relation.

selects all those tuples attribute.

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.

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 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 satisfying the following properties, which are designed to mimic the restriction of a function:

• then the composite

• is an open covering; and

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 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 for binary relations. These cases do not fit into the scheme of sheaves.

Anti-restriction

The domain anti-restriction (or domain subtraction⩤ Similarly, the range anti-restriction (or range subtraction⩥

See also

References