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\begin{document}

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\section*{locally constant function}

\hypertarget{locally_constant_functions}{}\section*{{Locally constant functions}}\label{locally_constant_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{pattern}{Pattern}\dotfill \pageref*{pattern} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A locally constant function is a [[function]] whose value never changes. This is a weaker concept than that of a [[constant function]], which has only one value. They correspond on a [[connected space]]. However, in general, a function may be locally constant but not constant, since it can take values on two distant components without the values' ever changing between them (since there is no path between them).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

If $X$ is a [[topological space]] and $Y$ is any [[set]], then a [[function]] $f$ from (the [[underlying set]] of) $X$ to $Y$ is \textbf{locally constant} if, for every element $a$ of $X$, $f$ is [[constant function|constant]] when restricted to some [[neighbourhood]] of $a$.

We have $Y$ here as a set; but in fact, $Y$ may be given any [[topological structure]]; then every locally constant function $f$ will become a locally constant [[continuous map]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{LocallyConstantFunctionIntoDiscreteSpace}\hypertarget{LocallyConstantFunctionIntoDiscreteSpace}{}
\textbf{(continuous function into [[discrete space]] is locally constant)}

A function into a [[discrete topological space]] is [[continuous function|continuous]] precisely if it is locally constant.

\end{example}
\hypertarget{pattern}{}\subsection*{{Pattern}}\label{pattern}

\begin{itemize}%
\item A \textbf{locally constant function} is a section of a [[constant sheaf]];


\item a [[locally constant sheaf]] is a section of a [[constant stack]];


\item a [[locally constant stack]] is a section of (\ldots{} and so on\ldots{})


\item a [[locally constant ∞-stack]] is a section of a [[constant ∞-stack]].



\end{itemize}
A locally constant sheaf / $\infty$-stack is also called a [[local system]].

[[!redirects locally constant functions]]



\end{document}