$C^1$ Map on an Open Subset of Euclidean Space

Let $n,m\in\mathbb{N}$. Let $U\subseteq \mathbb{R}^n$ be 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 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 Partial Derivative of a Coordinate Function $\frac{\partial f_j}{\partial x_i}(x)$ exists for every $x\in U$, with the function

$$x\mapsto \frac{\partial f_j}{\partial x_i}(x)$$

from $U$ to $\mathbb{R}$ also continuous at every point of $U$.