Composition of Continuous Euclidean Maps

Let $n,m,p\in\mathbb{N}$. Let $E\subseteq \mathbb{R}^n$, let $F\subseteq \mathbb{R}^m$, let $f:E\to F$, and let $g:F\to\mathbb{R}^p$. Let $a\in E$. Suppose that $f$ is continuous at $a$ and that $g$ is continuous at $f(a)$. Then the composition $g\circ f:E\to\mathbb{R}^p$ is continuous at $a$.