Let $V$ be an algebraic set of $\mathbb A^n$ (i.e. a Zariski-closed subset of $k^n$, for some field $k$). As usual a function $\phi : V \to k$ is regular at $p \in V$ if in some neighborhood of $V$ it is a rational function $g/h$, where $g$ and $h$ are in the coordinate ring $k[V]$. For a set $U$ open in $V$, we can then consider the ring of all open functions $\mathcal O_V(U)$.
Question:
Is it true that $\mathcal O_V(U)$ is the localization of the coordinate ring $k[V]$ at the (multiplicative) set of functions not vanishing on $U$?
I'm aware that the result is true if $U$ is a distinguished open set (i.e. $U = D(f)$ for some polynomial $f$), and false for schemes in general (Remark 4.1.5 in Vakil).
