> [!NOTE] Lemma (Composition of Continuous Functions in Continuous) > If $f:U \subset \mathbb{R}^n\to \mathbb{R}^k$ be [[Continuous maps|continuous]] at $p\in U$ with $f(U) \subset V$ and $g:V\subset \mathbb{R}^{m}$ is continuous at $f(p),$ then $g\circ f: U\to \mathbb{R}^m$ is continuous at $p.$ **Proof:** Let $x_j \subset U$ be a sequence which converges to $p$. Applying [[Equivalence of Continuous and Sequentially Continuous Functions Over Euclidean Spaces|equivalence of sequential continuity and continuity]] yields $\lim _{j \rightarrow \infty} f\left(x_j\right)=f(p)$. Then, by continuity of $g, \lim _{j \rightarrow \infty} g\left(f\left(x_j\right)\right)=g(f(p))$, i.e., $g \circ f: U \rightarrow \mathbb{R}^m$ is continuous at $p$. $\blacksquare$