Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, let $v \in \mathbb{R}^m$, and let $f \in C^2(U;\mathbb{R}^n)$. Suppose there exists $\varepsilon_0>0$ such that $a+tv \in U$ for every $t \in (-\varepsilon_0,\varepsilon_0)$. Then, as $t \to 0$ in $\mathbb{R}$,