Partial Derivative on a Euclidean Open Set

Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the real numbers with absolute value $|\cdot|$, let $f:U\to\mathbb{R}$, let $a=(a_1,\dots,a_n)\in U$, and let $i\in\{1,\dots,n\}$.

We say that the partial derivative of $f$ with respect to the $i$th variable exists at $a$, with value the real number $L$, if for every real $\varepsilon>0$ there exists a real $\delta>0$ such that every $h\in\mathbb{R}$ with $0<|h|<\delta$ satisfies $(a_1,\dots,a_{i-1},a_i+h,a_{i+1},\dots,a_n)\in U$ and

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

We say that the partial derivative of $f$ with respect to the $i$th variable exists at $a$ if it exists at $a$ with value $L$ for some real number $L$; any such $L$ is denoted by

$$\frac{\partial f}{\partial x_i}(a)\qquad\text{or}\qquad \partial_i f(a).$$