Zero-differential constancy theorem
= Zero-differential constancy theorem
{title2=$df\equiv0,\quad M\text{ connected}\Longrightarrow f\text{ constant}$}
A <smooth map> between <smooth manifolds> with zero differential is locally constant: each target coordinate function has zero <derivatives> on a sufficiently small source coordinate ball. Integration along straight segments proves constancy there. Its fibers are open and closed, so a <connected> source makes the map constant globally.