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[[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]]
== 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 '''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.
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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,
Further examples include the following:
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