Locally constant function: Difference between revisions

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{{Short description|Type of mathematical function}}

{{Unreferenced|date=December 2009}}

{{More citations needed|date=January 2024}}

[[File:Example of a local constant function with sgn(x).svg|thumb|The [[Signum function]] restricted to the ___domain <math>\R\setminus\{0\}</math> is locally constant.]]

{{redirect|Locally constant|the sheaf-theoretic term|locally constant sheaf}}

In [[mathematics]], a [[function (mathematics)|function]] ''f'' from a [[topological space]] ''A'' to a [[Set (mathematics)|set]] ''B'' is called '''locally constant''', if for every ''a'' in ''A'' there exists a [[neighborhood (topology)|neighborhood]] ''U'' of ''a'', such that ''f'' is constant on ''U''.

[[File:Example of a locallocally constant function with sgn(x).svg|thumb|The [[Signumsignum function]] restricted to the ___domain <math>\R\setminus\{0\}</math> is locally constant.]]

In [[mathematics]], a '''locally constant function''' is a [[functionFunction (mathematics)|function]] ''f'' from a [[topological space]] ''A'' tointo a [[Set (mathematics)|set]] ''B''with isthe calledproperty '''locallythat constant''',around ifevery forpoint everyof ''a''its in ''A''___domain, there exists asome [[neighborhoodNeighborhood (topology)|neighborhood]] ''U''of that point on which it [[Restriction of ''a'', suchfunction|restricts]] thatto ''f'' isa [[constant on ''U''function]].

== Definition ==

Every [[constant function]] is locally constant.

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>

Every locally constant function from the [[real number]]s '''R''' to '''R''' is constant, by the [[connected space|connectedness]] of '''R'''. But the function ''f'' from the [[rational number|rationals]] '''Q''' to '''R''', defined by ''f''(''x'') = 0 for ''x'' < [[Pi|&pi;]], and ''f''(''x'') = 1 for ''x'' > π, is locally constant (here we use the fact that π is [[irrational number|irrational]] and that therefore the two sets {''x''∈'''Q''' : ''x'' < π} and {''x''∈'''Q''' : ''x'' > π} are both [[open set|open]] in '''Q''').

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 ==

If ''f'' : ''A'' → ''B'' is locally constant, then it is constant on any [[connected space|connected component]] of ''A''. The converse is true for [[locally connected]] spaces (where the connected components are open).

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 [[connectedConnected space|connectedness]] of '''<math>\R'''.</math> But the function ''<math>f'' : \Q \to \R</math> from the [[rationalRational number|rationals]] '''<math>\Q'''</math> to '''<math>\R''',</math> defined by ''<math>f''(''x'') = 0 \text{ for ''} x'' < [[Pi|&\pi;]],</math> and ''<math>f''(''x'') = 1 \text{ for ''} x'' > π\pi,</math> is locally constant (here wethis useuses the fact that π<math>\pi</math> is [[irrationalIrrational number|irrational]] and that therefore the two sets <math>\{'' x''∈''' \in \Q''' : ''x'' < π\pi \}</math> and <math>\{'' x''∈''' \in \Q''' : ''x'' > π\pi \}</math> are both [[openOpen set|open]] in '''<math>\Q'''</math>).

If ''<math>f'' : ''A'' →\to ''B''</math> is locally constant, then it is constant on any [[connectedConnected space|connected component]] of ''<math>A''.</math> The converse is true for [[locally connected]] spaces, (wherewhich are spaces thewhose 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'' of\in ''X''</math> we can assign the [[cardinality]] of the [[Fiber (mathematics)|fiber]] ''p''<sup>−1</supmath>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 [[continuousContinuous 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 [[sheafSheaf (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 ''Z''<submath>''X''Z_X</submath>; described by means of ''{{em|stalks''}} we have stalk ''Z''<submath>''x''Z_x,</submath>, 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 isn'tis 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==

==Connection with sheaf theory==

{{Reflist}}

There are ''sheaves'' of locally constant functions on ''X''. To be more definite, the locally constant integer-valued functions on ''X'' form a [[sheaf (mathematics)|sheaf]] in the sense that for each open set ''U'' of ''X'' we can form the functions of this kind; and then verify that the sheaf ''axioms'' hold for this construction, giving us a sheaf of [[abelian group]]s (even [[commutative ring]]s). This sheaf could be written ''Z''_''X''; described by means of ''stalks'' we have stalk ''Z''_''x'', a copy of ''Z'' at ''x'', for each ''x'' in ''X''. This can be referred to a ''constant sheaf'', meaning exactly ''sheaf of locally constant functions'' taking their values in the (same) group. The typical sheaf of course isn't 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 ''locally'' look like such 'harmless' sheaves (near any ''x''), but from a global point of view exhibit some 'twisting'.

{{DEFAULTSORT:Locally Constant Function}}