Editing Locally constant function

{{Short description|Type of mathematical function}}
{{More citations needed|date=January 2024}}
{{redirect|Locally constant|the sheaf-theoretic term|locally constant sheaf}}
[[File:Example of a locally constant function with sgn(x).svg|thumb|The [[signum function]] restricted to the domain <math>\R\setminus\{0\}</math> is locally constant.]]
In [[mathematics]], a '''locally constant function''' is a [[Function (mathematics)|function]] from a [[topological space]] into a [[Set (mathematics)|set]] with the property that around every point of its domain, there exists some [[Neighborhood (topology)|neighborhood]] of that point on which it [[Restriction of a function|restricts]] to a [[constant function]].

== Definition ==

Let <math>f : X \to S</math> be a function from a [[topological space]] <math>X</math> into a [[Set (mathematics)|set]] <math>S.</math> 
If  <math>x \in X</math> then <math>f</math> is said to be '''locally constant at <math>x</math>''' if there exists a [[Neighborhood (topology)|neighborhood]] <math>U \subseteq X</math> of <math>x</math> such that <math>f</math> is constant on <math>U,</math> which by definition means that <math>f(u) = f(v)</math> for all <math>u, v \in U.</math> 
The function <math>f : X \to S</math> is called '''locally constant''' if it is locally constant at every point <math>x \in X</math> in its domain.

== Examples ==

Every [[constant function]] is locally constant. The converse will hold if its [[Domain of a function|domain]] is a [[connected space]].

Every locally constant function from the [[real number]]s <math>\R</math> to <math>\R</math> is constant, by the [[Connected space|connectedness]] of <math>\R.</math> But the function <math>f : \Q \to \R</math> from the [[Rational number|rationals]] <math>\Q</math> to <math>\R,</math> defined by <math>f(x) = 0 \text{ for } x < \pi,</math> and <math>f(x) = 1 \text{ for } x > \pi,</math> is locally constant (this uses the fact that <math>\pi</math> is [[Irrational number|irrational]] and that therefore the two sets <math>\{ x \in \Q : x < \pi \}</math> and <math>\{ x \in \Q : x > \pi \}</math> are both [[Open set|open]] in <math>\Q</math>).

If <math>f : A \to B</math> is locally constant, then it is constant on any [[Connected space|connected component]] of <math>A.</math> The converse is true for [[locally connected]] spaces, which are spaces whose connected components are open subsets.

Further examples include the following:
* Given a [[covering map]] <math>p : C \to X,</math> then to each point <math>x \in X</math> we can assign the [[cardinality]] of the [[Fiber (mathematics)|fiber]] <math>p^{-1}(x)</math> over <math>x</math>; this assignment is locally constant.
* A map from a topological space <math>A</math> to a [[discrete space]] <math>B</math> is [[Continuous function (topology)|continuous]] if and only if it is locally constant.

== Connection with sheaf theory ==

There are {{em|sheaves}} of locally constant functions on <math>X.</math> To be more definite, the locally constant integer-valued functions on <math>X</math> form a [[Sheaf (mathematics)|sheaf]] in the sense that for each open set <math>U</math> of <math>X</math> we can form the functions of this kind; and then verify that the sheaf {{em|axioms}} hold for this construction, giving us a sheaf of [[abelian group]]s (even [[commutative ring]]s).<ref>{{cite book |last1=Hartshorne |first1=Robin |title=Algebraic Geometry |date=1977 |publisher=Springer |page=62}}</ref> This sheaf could be written <math>Z_X</math>; described by means of {{em|stalks}} we have stalk <math>Z_x,</math> a copy of <math>Z</math> at <math>x,</math> for each <math>x \in X.</math> This can be referred to a {{em|constant sheaf}}, meaning exactly {{em|sheaf of locally constant functions}} taking their values in the (same) group. The typical sheaf of course is not constant in this way; but the construction is useful in linking up [[sheaf cohomology]] with [[homology theory]], and in logical applications of sheaves. The idea of [[local coefficient system]] is that we can have a theory of sheaves that {{em|locally}} look like such 'harmless' sheaves (near any <math>x</math>), but from a global point of view exhibit some 'twisting'.

== See also ==

* {{annotated link|Liouville's theorem (complex analysis)}}
* [[Locally constant sheaf]]

==References==
{{Reflist}}

{{DEFAULTSORT:Locally Constant Function}}
[[Category:Sheaf theory]]