Locally constant function

The restricted sign function is locally constant

{\ displaystyle \ mathbb {R} \ setminus \ {0 \}}

In mathematics , a function from a topological space into a set is called locally constant if for each there is a neighborhood of on which is constant . ${\ displaystyle f \ colon T \ to M}$ ${\ displaystyle T}$ ${\ displaystyle M}$ ${\ displaystyle x \ in T}$ ${\ displaystyle U}$ ${\ displaystyle x}$ ${\ displaystyle f}$

properties

Every constant function is also locally constant.

Each locally constant function of in an arbitrary amount is constant, since contiguous and not by at least two disjoint open sets to overlap is. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle M}$ ${\ displaystyle \ mathbb {R}}$

Every locally constant holomorphic function from an open set into the complex numbers is constant if a domain is, that is , if it is connected. ${\ displaystyle f \ colon M \ to \ mathbb {C}}$ ${\ displaystyle M}$ ${\ displaystyle M}$

In general, every locally constant function is constant on every connected component ; the converse also applies to locally connected spaces .

A mapping from a topological space into a discrete space is continuous if and only if it is locally constant. ${\ displaystyle f \ colon T \ to D}$ ${\ displaystyle T}$ ${\ displaystyle D}$

Every mapping from a discrete space to any topological space is locally constant.

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The set of locally constant functions in a space naturally form a sheaf of commutative rings .

Examples

The function defined by for and for is locally constant. (This implies that it is irrational , since such and are open sets that cover.) ${\ displaystyle f \ colon \ mathbb {Q} \ to \ mathbb {Q}}$ ${\ displaystyle f (x) = 0}$ ${\ displaystyle x <\ pi}$ ${\ displaystyle f (x) = 1}$ ${\ displaystyle x> \ pi}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ {x \ mid x <\ pi \}}$ ${\ displaystyle \ {x \ mid x> \ pi \}}$ ${\ displaystyle \ mathbb {Q}}$