Given a subset $U\subseteq \mathbb{R}^2$, let us say that a point $p$ lying in the boundary of $U$ is a simple boundary point if there exists a continuous curve $γ:(-1,1)→\mathbb{R}^2$, such that
$γ(0)=p$,
$γ(t)∈\text{int}(U)$, for all $t∈(0,1)$, and
$γ(t)∈\text{ext}(U)$, for all $t∈(-1,0)$,
where int and ext refer, respectively, to interior and exterior.
Assuming that $U$ is a nonempty open set, and that its exterior is nonempty, can we guarantee the existence of simple boundary points? If so, can we also prove that the set of simple boundary points is large in any reasonable sense?
PS. It is not hard to see that there is a point in the boundary of the Warsaw circle which is not simple.