Uniform Bounds in D-Minimal Structures

Let $\mathcal{R}$ be an expansion of the real field such that every subset of $\mathbb{R}$ definable in $\mathcal{R}$ either has interior or is a finite union of discrete sets. Answering a question by Chris Miller, we show that for every $n\in \mathbb{N}$ and every definable subset $A\subseteq {\mathbb{R}}^{n+1}$ there is $N\in \mathbb{N}$ such that for all $x\in {\mathbb{R}}^n$ either $A_x$ has interior or is the union of $N$ discrete sets.