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B doesn't have to be a topological space; more examples, facts |
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[[de:Lokal konstante Funktion]]
In [[mathematics]], a [[function]] ''f'' from a [[topological space]] ''A'' to a [[set]] ''B'' is called '''locally constant''', [[iff]] for every ''a'' in ''A'' there exists a [[neighborhood (topology)|neighborhood]] ''U'' of ''a'', such that ''f'' is constant on ''U''.
Every constant function is locally constant.
Every locally constant function from the [[real number]]s '''R''' to '''R''' is constant. 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'''.
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