Hurwitz' theorem says the following. Say $(f_n)$ is a sequence of a holomorphic functions (on a domain $\Omega$ in $\mathbb{C}$) converging uniformly on compact subsets to a non-constant holomorphic function $f$. Then, if $f$ has a zero at $z \in \Omega$, for $\rho<0$ sufficiently small, and $K \in \mathbb{N}$ sufficiently large, each $f_n, n \geq k,$ will have a zero in the disc $D(z, \rho)$...moreover these zeros converge to $z$ as $k \to \infty$.
In rough terms, Hurwitz' theorem says that if we perturb a holomorphic function slightly, we only move the zeros slightly.
Does anyone know if the following is true (and if so, may I be directed to a proof of the same?): if $F: [0,1] \times \Omega \to \mathbb{C}$ is a continuous family of holomorphic functions (i.e., $F(t, \cdot)$ is holomorphic), and, for some $s \in [0,1]$, $F(s, \cdot)$ has a zero at $\zeta_s$, then there is a neighbourhood of $V \subset [0,1]$ of $s$ with a continuous function $V \to \Omega, t \mapsto \zeta_t,$ such that $F(t, \zeta_t)=0$?
It seems like it must be true!