Locally constant function: Difference between revisions

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== 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>).