Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.


Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x = f(0)\ne f(1)$?


That is, does then there always exist a nontrvial continuous curve starting at $x$ and staying outside $U$?



On the boundary of a simply connected set
Here's a variant of Fernando Muro's construction.


Let $X$ be the closure in $\mathbb R^2$of the graph of the function
$g(x):=x\sin\big(1/\sin(1/x)\big)$ defined on $[0,+\infty)\setminus\{1/n\pi:n\in \mathbb Z\_+\}$. Let $U:=\mathbb R^2\setminus X$, again a simply connected open set. The function $g$ has a set of essential discontinuities $\{1/n\pi:n\in \mathbb Z\_+\}$ that accumulates at $0$, but it is continuous at $0$, so $X$ has no points with $0$ first coordinate but $(0,0)$, and $(0,0)$ can be connected by continuous arcs to no other point of $X$ .


