More specifically, let $B$ be a open ball and $C, D$ be open disjoint sets in $\mathbb{R}^n$, $n>1$. Suppose that $B\cap C\neq\emptyset$ and $B\cap D\neq\emptyset$, furthermore, $B\subset \bar{C}\cup\bar{D}$. Is there at least one path in $B\cap\partial C$?

Edit: for what i need, the statement actually can be a little less strong: is there some path $\varphi$ in $B$ such that $\varphi\cup\partial C$ has uncontable many points?