Here are two definitions of continuity of preferences. Denote the (weak) preference relation by ≽. We assume completeness, reflexivity and transitivity. Assume non-satiation or strict monotonicity only if necessary (and if you do so, please mention).
Definition 1 (Standard): If $(x_n)$ and $(y_n)$ are two sequences such that $x_n \to x$ and $y_n \to y$, then if $x_n ≽ y_n$ for all $n$, we have $x ≽ y$.
Definition 2 (Varian/NS): If $x \succ y$ and $z$ is "sufficiently close" to $x$, then $z \succ y$.
Can we prove that these two are equivalent?
Here's an attempt.
Proof that Def. 2 implies Def. 1: Suppose not. Assume Varian's definition. Then if we have $(x_n)$ and $(y_n)$ with the given criteria, the result would be $y ≽ x$ (due to completeness). Since (for all $n$) $y_n$ is sufficiently close to $y$, by Def. 2, $y_n ≽ y$ which is not necessarily true in general. Thus, Def. 2 implies Def. 1.
Can we show that Def. 1 implies Def. 2? It seems like we need additional conditions, but I can't figure out what all is needed, if at all.
Note: "Sufficiently close" is an informal terminology used by Varian, so you can treat it this way - if $x$ and $z$ are sufficiently close, that is, if $z$ lies in the $\epsilon$-ball of $x$ (written $B(x,\epsilon)$), then $\epsilon > 0$ can be set as small as you wish.
Edit: Varian/Nicholson_Snyder used strict preferences which has now been incorporated.