Partial Derivative of a Coordinate Function

Let $n,m\in\mathbb{N}$. Let $U\subseteq \mathbb{R}^n$ be open, let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$, let $a=(a_1,\dots,a_n)\in U$, and fix indices $i\in\{1,\dots,n\}$ and $j\in\{1,\dots,m\}$. We say that the partial derivative of the $j$th coordinate function of $f$ with respect to the $i$th variable exists at $a$ if there exists a real number $L$ such that for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $h\in\mathbb{R}$ satisfies $0<|h|<\delta$, one has

$$\left|\frac{f_j(a_1,\dots,a_{i-1},a_i+h,a_{i+1},\dots,a_n)-f_j(a_1,\dots,a_n)}{h}-L\right|<\varepsilon.$$

In that case $L$ is called the partial derivative of $f_j$ with respect to $x_i$ at $a$ and is denoted by

$$\frac{\partial f_j}{\partial x_i}(a).$$