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For other uses, see Restriction (disambiguation).

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Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable | |||||||||||||||||||||||||||||

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Restriction · Composition · λ · Inverse | |||||||||||||||||||||||||||||

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In mathematics, the notion of **restriction of a function** is defined as follows:

If *f* : *E* → *F* is a function from E to F, and A is a subset of E, then the **restriction of f to A** is the function

- having the graph .

(In rough words, it is "the same function", but only defined on .)

More generally, the restriction (or **domain restriction** or **left-restriction**) *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 **right-restriction** or **range restriction** *R* ▷ *B*. Indeed, one could define a restriction to n-ary 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.^{[clarification needed]}

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 (*E* \ *A*) ◁ *R*; 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 *R* ▷ (*F* \ *B*); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.

- The restriction of the non-injective function to is the injection .
- The inclusion map of a set A into a superset E of A is the restriction of the identity function on E to A.