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 \reftext{def:continuous-map-at-point-euclidean-2026a}{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 \reftext{def:continuous-at-point-c54-2026b}{continuity for real-valued functions}.