Coordinatewise Characterization of Continuity for Euclidean Maps

Let $n,m\in\mathbb{N}$. Let $E\subseteq \mathbb{R}^n$, let $f=(f_1,\dots,f_m):E\to\mathbb{R}^m$, and let $a\in E$. Then the following are equivalent.

1. The map $f$ is continuous at $a$.
2. For every index $j\in\{1,\dots,m\}$, the coordinate function $f_j:E\to\mathbb{R}$ is continuous at $a$ in the sense of continuity for real-valued functions.