Editing Locally constant function (section)

== 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.