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 of class $C^1$ on $U$ if each coordinate function $f_j:U\to\mathbb{R}$ is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of $U$}, and if for every $j\in\{1,\dots,m\}$ and every $i\in\{1,\dots,n\}$ the partial derivative \ref{def:partial-derivative-coordinate-map-2026a} $\frac{\partial f_j}{\partial x_i}(x)$ exists for every $x\in U$, with the function

from $U$ to $\mathbb{R}$ also \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of $U$}.