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 *U* of *a*, such that *f* is constant on *U*.

Every constant function is locally constant.

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* > π, is locally constant (here we use the fact that π is irrational and that therefore the two sets {*x*∈**Q** : *x* < π} and {*x*∈**Q** : *x* > π} are both open in **Q**.

Generally speaking, if *f* : *A* → *B* is locally constant, then it is constant on any connected component of *A*. The converse is true for locally connected spaces (where the connected components are open).

Further examples include the following:

- Given a covering
*p*:*C*→*X*, then to each point*x*of*X*we can assign the cardinality of the fibre*p*^{-1}(*x*) over*x*; this assignment is locally constant. - A map from the topological space
*A*to a discrete space*B*is continuous if and only if it is locally constant.