Restriction (mathematics)#Extension of a function

File:Inverse square graph.svg. If we restrict $x^2$ to the non-negative real numbers, then it does have an inverse function, known as the square root of $x.$ ]]

In mathematics, the restriction of a function $f$ is a new function, denoted $f\vert_A$ or $f {\restriction_A},$ 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

{{Cite book|last=Stoll|first=Robert|title=Sets, Logic and Axiomatic Theories|publisher=W. H. Freeman and Company|date=1974|location=San Francisco|pages=[36]|edition=2nd|isbn=0-7167-0457-9|url=https://archive.org/details/setslogicaxiomat0000stol/page/5}}

${f|}_A : A \to F$

given by ${f|}_A(x) = f(x)$ 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,

: $G({f|}_A) = \{ (x,f(x))\in G(f) : x\in A \} = G(f)\cap (A\times F),$

where the pairs $(x,f(x))$ represent ordered pairs in the graph $G.$

=Extensions=

A function $F$ is said to be an {{visible anchor|Extension of a function|text=extension}} of another function $f$ if whenever $x$ is in the domain of $f$ then $x$ is also in the domain of $F$ and $f(x) = F(x).$

That is, if $\operatorname{domain} f \subseteq \operatorname{domain} F$ and $F\big\vert_{\operatorname{domain} f} = f.$

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

The restriction of the non-injective function $f: \mathbb{R} \to \mathbb{R}, \ x \mapsto x^2$ to the domain $\mathbb{R}_{+} = [0,\infty)$ is the injection $f:\mathbb{R}_+ \to \mathbb{R}, \ x \mapsto x^2.$
The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: ${\Gamma|}_{\mathbb{Z}^+}\!(n) = (n-1)!$

Properties of restrictions

Restricting a function $f:X\rightarrow Y$ to its entire domain $X$ gives back the original function, that is, $f|_X = f.$
Restricting a function twice is the same as restricting it once, that is, if $A \subseteq B \subseteq \operatorname{dom} f,$ then $\left(f|_B\right)|_A = f|_A.$
The restriction of the identity function on a set $X$ to a subset $A$ of $X$ is just the inclusion map from $A$ into $X.$ {{cite book|author-link=Paul Halmos|last=Halmos|first=Paul|title=Naive Set Theory|location=Princeton, NJ|publisher=D. Van Nostrand|year=1960}} Reprinted by Springer-Verlag, New York, 1974. {{isbn|0-387-90092-6}} (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. {{isbn|978-1-61427-131-4}} (Paperback edition).
The restriction of a continuous function is continuous.{{cite book|last=Munkres|first=James R.|title=Topology|edition=2nd|location=Upper Saddle River|publisher=Prentice Hall|year=2000|isbn=0-13-181629-2}}{{cite book|last=Adams|first=Colin Conrad|first2=Robert David|last2=Franzosa|title=Introduction to Topology: Pure and Applied|publisher=Pearson Prentice Hall|year=2008|isbn=978-0-13-184869-6}}

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 $x^2 = (-x)^2$ for any $x \in \R.$ However, the function becomes one-to-one if we restrict to the domain $\R_{\geq 0} = [0, \infty),$ in which case

$f^{-1}(y) = \sqrt{y} .$

(If we instead restrict to the domain $(-\infty, 0],$ 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

$\sigma_{a \theta b}(R)$ or $\sigma_{a \theta v}(R)$ where:

$a$ and $b$ are attribute names,
$\theta$ is a binary operation in the set $\{<, \leq, =, \neq, \geq, >\},$
$v$ is a value constant,
$R$ is a relation.

The selection $\sigma_{a \theta b}(R)$ selects all those tuples in $R$ for which $\theta$ holds between the $a$ and the $b$ attribute.

The selection $\sigma_{a \theta v}(R)$ 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 $A = X \cup Y,$ 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 $V\subseteq U,$ then there is a morphism $\operatorname{res}_{V,U} : F(U) \to F(V)$ satisfying the following properties, which are designed to mimic the restriction of a function:

For every open set $U$ of $X,$ the restriction morphism $\operatorname{res}_{U,U} : F(U) \to F(U)$ is the identity morphism on $F(U).$
If we have three open sets $W \subseteq V \subseteq U,$ then the composite $\operatorname{res}_{W,V} \circ \operatorname{res}_{V,U} = \operatorname{res}_{W,U}.$
(Locality) If $\left(U_i\right)$ is an open covering of an open set $U,$ and if $s, t \in F(U)$ are such that $s\big\vert_{U_i} = t\big\vert_{U_i}$ for each set $U_i$ of the covering, then $s = t$ ; and
(Gluing) If $\left(U_i\right)$ is an open covering of an open set $U,$ and if for each $i$ a section $x_i \in F\left(U_i\right)$ 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: $s_i\big\vert_{U_i \cap U_j} = s_j\big\vert_{U_i \cap U_j},$ then there is a section $s \in F(U)$ such that $s\big\vert_{U_i} = s_i$ for each $i.$

The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf.

Left- and right-restriction

More generally, the restriction (or domain restriction or left-restriction) $A \triangleleft R$ of a binary relation $R$ between $E$ and $F$ may be defined as a relation having domain $A,$ codomain $F$ and graph $G(A \triangleleft R) = \{(x, y) \in F(R) : x \in A\}.$ Similarly, one can define a right-restriction or range restriction $R \triangleright B.$ 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.{{clarify|date=July 2013}}

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 $(E \setminus A) \triangleleft R$ ; it removes all elements of $A$ from the domain $E.$ It is sometimes denoted $A$ ⩤ $R.$ Dunne, S. and Stoddart, Bill Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 5–7, 2006, Revised Selected ... Computer Science and General Issues). Springer (2006) Similarly, the range anti-restriction (or range subtraction) of a function or binary relation $R$ by a set $B$ is defined as $R \triangleright (F \setminus B)$ ; it removes all elements of $B$ from the codomain $F.$ It is sometimes denoted $R$ ⩥ $B.$

References

Category:Sheaf theory