Every locally constant continuous 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|π]], 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 ''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).
