Let $k,n$ be \reftext{def:natural-numbers-2026a}{natural numbers} with $k\ge 1$, let $U$ be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$, and let $f,g:U\to\mathbb{R}$ be maps of class \reftext{def:ck-map-euclidean-open-set-2026b}{$C^k$} on $U$.

(i) The pointwise product $(fg):U\to\mathbb{R}$, defined by $(fg)(x)=f(x)\,g(x)$ for $x\in U$, is of class \reftext{def:ck-map-euclidean-open-set-2026b}{$C^k$} on $U$.

(ii) If $g(x)\ne 0$ for every $x\in U$, then the pointwise quotient $f/g:U\to\mathbb{R}$, defined by $(f/g)(x)=f(x)/g(x)$ for $x\in U$, is of class \reftext{def:ck-map-euclidean-open-set-2026b}{$C^k$} on $U$.