Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In what sense should the $F_n$ converge to $F$, and what additional conditions should be placed on them, to ensure that (for large enough $n$) we can find a sequence of $\mathbf{x}_n$ with $F_n(\mathbf{x}_n)=0$ and $\mathbf{x}_n$ converging to $\mathbf{x}$?

Apologies for the elementary question. I know this must be a very standard result but I can't seem to find it. Can anyone point me to the right reference?