One-types over the natural numbers in the rational order ($S_1^{(\mathbb{Q},<)}(\mathbb{N})$)

Quantifier elimination for dense linear orders shows that the one-types over $\mathbb{N}\subset \mathbb{Q}$ are determined by cuts: equality to a natural number, one of the intervals between consecutive natural numbers, the ray below zero, or the cut above every natural number. Only the last type is non-isolated.