Let $X$ be a complex variety and $D \subset X$ an hyperusrface. We say that a function $$f: X-D \to \mathbb{C}$$ is meromorphic along $D$ if for every $p \in D$ there exixsts $V_p \ni p $ open subset such that $$f|_{V_p}=\dfrac{s}{g}$$ , $s,g $ holomorphic in $V_p$.
Is it true that for every $p \in D$ there exists an open subset such that every such function is written as $\dfrac{s}{h}$ where $h$ is a local defining equation for $D$ , i.e $D \cap U_p=\{h=0\}$ with $U_p \ni p$ open subset