Let $k,n,m$ 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 $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$.

We say that $f$ is of class $C^k$ on $U$ if for every multi-index $\alpha$ of length $n$ with \reftext{def:order-factorial-multi-index-2026a}{order} $|\alpha|\le k$ and every index $j\in\{1,\dots,m\}$, the partial derivative $\partial^\alpha f_j$ of \reftext{def:partial-derivative-order-alpha-2026a}{order $\alpha$} exists on $U$, and the resulting function

is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of $U$}.

A real-valued map $f:U\to\mathbb{R}$ is of class $C^k$ if it is of class $C^k$ as a map from $U$ to $\mathbb{R}^1$.