In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.

Definition

Let f : X \to S be a function from a topological space X into a set S. If x \in X then f is said to be locally constant at x if there exists a neighborhood of x such that f is constant on U, which by definition means that f(u) = f(v) for all u, v \in U. The function f : X \to S is called locally constant if it is locally constant at every point x \in X in its domain.

Examples

Every constant function is locally constant. The converse will hold if its domain is a connected space. Every locally constant function from the real numbers \R to \R is constant, by the connectedness of \R. But the function from the rationals \Q to \R, defined by and is locally constant (this uses the fact that \pi is irrational and that therefore the two sets and are both open in \Q). If f : A \to B is locally constant, then it is constant on any connected component of A. The converse is true for locally connected spaces, which are spaces whose connected components are open subsets. Further examples include the following:

Connection with sheaf theory

There are 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 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 we have stalk Z_x, a copy of Z at x, for each x \in X. This can be referred to a, meaning exactly taking their values in the (same) group. The typical sheaf of course is not 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 look like such 'harmless' sheaves (near any x), but from a global point of view exhibit some 'twisting'.