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 neighborhood 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 < π, and f(x) = 1 for x > π, is locally constant.