Continuity at a Point for Maps Between Euclidean Spaces

Let $n,m\in\mathbb{N}$. Let $E\subseteq \mathbb{R}^n$, let $f:E\to\mathbb{R}^m$, let $a\in E$, and write $f=(f_1,\dots,f_m)$. We say that $f$ is continuous at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for every point $x=(x_1,\dots,x_n)\in E$, if

$$\sum_{i=1}^n (x_i-a_i)^2<\delta^2,$$

then

$$\sum_{j=1}^m \bigl(f_j(x)-f_j(a)\bigr)^2<\varepsilon^2.$$