The way that I imagine local connectedness in the plane is the following "theorem".
"Theorem": A set $E\subset\mathbb{C}$ is locally connected at a point $x\in E$ if, for any $\epsilon>0$, there is a $\delta>0$ such that if $y\in E$ with $|x-y|<\delta$, then there is a path in $E$ from $x$ to $y$ which is contained in the $\epsilon$-ball around $x$.
I am sure this is a naive way of thinking about it. I am interested in finding an example which will cure me of this way of thinking. That is, a set $E$ which is locally connected at some point $x\in E$, but that does not satisfy the property described in the "Theorem".