# Locally constant function

In mathematics, a function *f* from a topological space *A* to a set *B* is called **locally constant**, if 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, by the connectedness of **R**. 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 (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**).

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

## Connection with sheaf theory

There are *sheaves* of locally constant functions on *X*. To be more definite, the locally constant integer-valued functions on *X* form a sheaf in the sense that for each open set *U* of *X* we can form the functions of this kind; and then verify that the sheaf *axioms* hold for this construction, giving us a sheaf of abelian groups (even commutative rings). This sheaf could be written *Z*_{X}; described by means of *stalks* we have stalk *Z*_{x}, a copy of *Z* at *x*, for each *x* in *X*. This can be referred to a *constant sheaf*, meaning exactly *sheaf of locally constant functions* taking their values in the (same) group. The typical sheaf of course isn't constant in this way; but the construction is useful in linking up sheaf cohomology with homology theory, and in logical applications of sheaves. The idea of local coefficient system is that we can have a theory of sheaves that *locally* look like such 'harmless' sheaves (near any *x*), but from a global point of view exhibit some 'twisting'.