Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}, let $x_0$ be a point of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$, and let $r,s\in\mathbb{R}$ with $0<r<s$. Then there exists a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} $\chi:\mathbb{R}^n\to\mathbb{R}$ such that, with $d$ denoting the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} on $\mathbb{R}^n$:

1. $0\le\chi(x)\le 1$ for every $x\in\mathbb{R}^n$;
2. $\chi(x)=1$ for every $x\in\mathbb{R}^n$ with $d(x,x_0)\le r$;
3. $\chi(x)=0$ for every $x\in\mathbb{R}^n$ with $d(x,x_0)\ge s$.