Let $n,m\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let $F=(F_1,\dots,F_m):U\to\mathbb{R}^m$. We say that $F$ is smooth on $U$ if for every multi-index $\alpha$ of length $n$ and every index $j\in\{1,\dots,m\}$, the partial derivative of $F_j$ of order $\alpha$ exists on $U$ in the sense of \ref{def:partial-derivative-order-alpha-2026a}, 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 smooth if it is smooth as a map from $U$ to $\mathbb{R}^1$.