Let $\mathcal{ R}$ be an $\mathrm{ NIP}$ expansion of $(\mathbb{ R},<,+)$ by closed subsets of $\mathbb{ R}^n$ and continuous functions $f : \mathbb{ R}^m \to \mathbb{ R}^n$. Then $\mathcal{ R}$ is generically locally o-minimal. It follows that if $X \subseteq \mathbb{ R}^n$ is definable in $\mathcal{ R}$ then the $C^k$-points of $X$ are dense in $X$ for any $k \geq 0$. This follows from a more general theorem on $\mathrm{ NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and $\bigwedge$-definable. We also show that $\mathcal{ R}$ is strongly dependent if and only if $\mathcal{ R}$ is either o-minimal or $(\mathbb{ R},<,+,\alpha\mathbb{ Z})$-minimal for some $\alpha > 0$.
Categories: math.LO