Coordinatewise Criterion for Continuity of a Map Between Euclidean Spaces

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, let $E$ be a subset of Euclidean space $\mathbb{R}^n$, let $f=(f_1,\dots,f_m):E\to\mathbb{R}^m$ with coordinate functions $f_j:E\to\mathbb{R}$, and let $a\in E$. Each $f_j$ is regarded as a map from $E$ into $\mathbb{R}^1$ with $f_j$ as its single coordinate function.

Then $f$ is continuous at $a$ if and only if $f_j$ is continuous at $a$ for every $j\in\{1,\dots,m\}$.