Question on skyscraper sheaf and direct limit.

Question on skyscraper sheaf and direct limit.

Let $X$ be a topological space, $i:Z\rightarrow X$ the inclusion and
$\mathcal{F}$ a sheaf on $Z$.
Assume that $Z$ is locally connected and that $\mathcal{F}$ is a constant
sheaf with value $E$ where $E$ is some set. Show that
$i_{*}(\mathcal{F})_{x}=E$ for all $x\in\overline{Z}$.
Question
Because $\mathcal{F}$ is a constant sheaf on $Z$, so naturally for all
$x\in Z$, $i_{*}(\mathcal{F})_{x}=\mathcal{F}_{x}=E$.
The question is why would this be true even for $x\in\overline{Z}$? Let's
say in some cases where $\overline{Z}-Z$ is not empty, and let $x$ be an
element in such a set. Then
$\displaystyle
i_{*}(\mathcal{F})_{x}=\lim_{\rightarrow}i_{*}(\mathcal{F})(U)$
where the limit is taken over open sets containing $x$. I tried to reason
out using the fact that $\mathcal{F}(U)$ may be seen as a set of
continuous functions from $U$ to $E$ where $E$ is given a discrete
topology, but I was stuck.
I went to find other sources, but they just say "since every open
neighbourhood of $x$ has non-empty intersection with $Z-\{x\}$, so the
direct limit lies inside $E$"
I would like to know why this should be true. Or rather, we know that the
direct limit should be a set, but why should it be $E$ just because every
neighbourhood of $x$ has non-empty intersection with $Z-\{x\}$?
Thanks!