> [!NOTE] Lemma (Statement) > Write $f: U \subset \mathbb{R}^n \to \mathbb{R}^k$ as $x \mapsto f(x)=(f_{1}(x),\dots,f_{k}(x)).$ $f$ is [[Continuous maps|continuous]] at $p\in U$ if, and only if, for all $i\in \{ 1,\dots,k \}$, $f_{i}:U\to \mathbb{R}$ is continuous at $p$. ###### Proof of statement: Follows from [[Equivalence of Continuous and Sequentially Continuous Functions Over Euclidean Spaces|sequential definition of continuity]] and [[Equivalence of Component-wise Convergence and Convergence for Sequences in Euclidean Space|and the equivalence of convergence and component-wise convergence for sequences in]] $\mathbb{R}^n$. $\blacksquare$