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Inmathematics, alocally constant function is afunction from atopological space into aset with the property that around every point of its domain, there exists someneighborhood of that point on which itrestricts to aconstant function.

Let${\displaystyle f:X\to S}$ be a function from atopological space${\displaystyle X}$ into aset${\displaystyle S.}$ If${\displaystyle x\in X}$ then${\displaystyle f}$ is said to belocally constant at${\displaystyle x}$ if there exists aneighborhood${\displaystyle U\subseteq X}$ of${\displaystyle x}$ such that${\displaystyle f}$ is constant on${\displaystyle U,}$ which by definition means that${\displaystyle f(u)=f(v)}$ for all${\displaystyle u,v\in U.}$ The function${\displaystyle f:X\to S}$ is calledlocally constant if it is locally constant at every point${\displaystyle x\in X}$ in its domain.

Everyconstant function is locally constant. The converse will hold if itsdomain is aconnected space.

Every locally constant function from thereal numbers${\displaystyle \mathbb{R}}$ to${\displaystyle \mathbb{R}}$ is constant, by theconnectedness of${\displaystyle \mathbb{R}.}$ But the function${\displaystyle f:\mathbb{Q}\to \mathbb{R}}$ from therationals${\displaystyle \mathbb{Q}}$ to${\displaystyle \mathbb{R},}$ defined by and${\displaystyle f(x)=1\text{ for }x>\pi ,}$ is locally constant (this uses the fact that${\displaystyle \pi }$ isirrational and that therefore the two sets and${\displaystyle \{x\in \mathbb{Q}:x>\pi \}}$ are bothopen in${\displaystyle \mathbb{Q}}$).

If${\displaystyle f:A\to B}$ is locally constant, then it is constant on anyconnected component of${\displaystyle A.}$ The converse is true forlocally connected spaces, which are spaces whose connected components are open subsets.

Further examples include the following:

• Given acovering map${\displaystyle p:C\to X,}$ then to each point${\displaystyle x\in X}$ we can assign thecardinality of thefiber${\displaystyle p^{-1}(x)}$ over${\displaystyle x}$; this assignment is locally constant.
• A map from a topological space${\displaystyle A}$ to adiscrete space${\displaystyle B}$ iscontinuous if and only if it is locally constant.

There aresheaves of locally constant functions on${\displaystyle X.}$ To be more definite, the locally constant integer-valued functions on${\displaystyle X}$ form asheaf in the sense that for each open set${\displaystyle U}$ of${\displaystyle X}$ we can form the functions of this kind; and then verify that the sheafaxioms hold for this construction, giving us a sheaf ofabelian groups (evencommutative rings).^[1] This sheaf could be written${\displaystyle Z_X}$; described by means ofstalks we have stalk${\displaystyle Z_x,}$ a copy of${\displaystyle Z}$ at${\displaystyle x,}$ for each${\displaystyle x\in X.}$ This can be referred to aconstant sheaf, meaning exactlysheaf 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 upsheaf cohomology withhomology theory, and in logical applications of sheaves. The idea oflocal coefficient system is that we can have a theory of sheaves thatlocally look like such 'harmless' sheaves (near any${\displaystyle x}$), but from a global point of view exhibit some 'twisting'.

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

• Sheaf theory

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