Before posting this question,I just asked a similar question:a question about connected open sets in $R^2$. I got several nice answers.Now I want to ask:

Let $U$ be a nonempty connected open set in $\mathbb{R}^2$ and $U\not=\mathbb{R}^2$.I want to ask if there must exist an open ball $B\subset \mathbb{R}^2$ such that $B\not\subset U$ and $B\cap U$ is a nonempty connected open set.