Let $A$ be the set of points $(x,y)\in\mathbb{R}^2$ with $x=0,|y|\leq 1$, and let $B$ be the set with $x>0,y=\sin 1/x$. Is $A\cup B$ connected?
From the picture, I think it should be connected, because the points in $B$ oscillates between $y=-1$ and $y=1$ as $x$ approaches $0$, and the points in $A$ cover all of the range $[-1,1]$ for $x=0$. However, to prove it rigorously, I must show that there doesn't exist disjoint, nonempty, open sets $C, D$ such that $A\cup B=C\cup D$. How can I show that?
