Locally constant function

A function f from a topological space A to a topological space B is called locally constant, iff for every a in A there exists an environment U of a, such that f is constant in U.

Every locally constant function from the real numbers R to R is constant. But the function f from the rationals Q to R, defined by f(x)=0 for x<Pi, and f(x)=1 for x>Pi, is locally constant.