Locally constant function

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.

The signum function restricted to the domain $\mathbb {R} \setminus \{0\}$ is locally constant.

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 $U\subseteq X$ 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 $\mathbb {R}$ to $\mathbb {R}$ is constant, by the connectedness of $\mathbb {R} .$ But the function $f:\mathbb {Q} \to \mathbb {R}$ from the rationals $\mathbb {Q}$ to $\mathbb {R} ,$ defined by $f(x)=0{\text{ for }}x<\pi ,$ and $f(x)=1{\text{ for }}x>\pi ,$ is locally constant (this uses the fact that $\pi$ is irrational and that therefore the two sets $\{x\in \mathbb {Q} :x<\pi \}$ and $\{x\in \mathbb {Q} :x>\pi \}$ are both open in $\mathbb {Q}$).

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:

• Given a covering map $p:C\to X,$ then to each point $x\in X$ 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 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 locally look like such 'harmless' sheaves (near any $x$), but from a global point of view exhibit some 'twisting'.

See also

• Liouville's theorem (complex analysis) – Theorem in complex analysis
• Locally constant sheaf