Before posting this question,I just asked a similar question:http://mathoverflow.net/questions/188207/a-question-about-connected-open-sets-in-r2.
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.