I'm trying to solve exercise 2.14c in Gortz-Wedhorn's book on algebraic geometry, and it looks to me like it's wrong.
Here's the statement. Let $X$ be a topological space and $i:Z \rightarrow X$ the inclusion of a subspace. Assume that $Z$ is locally connected and that $\mathcal{F}$ is a constant sheaf on $Z$ with value $E$, where $E$ is some set. Show that $i_{\ast}(\mathcal{F})_x = E$ for all $x \in \overline{Z}$.
Isn't the following a counterexample? Let $X = \mathbb{R}$ (with the usual topology) and let $Z = \mathbb{R} \setminus \{0\}$. Let $\mathcal{F}$ be the sheaf of constant $\mathbb{R}$-valued functions on $Z$. Then it looks to me like $i_{\ast}(\mathcal{F})_0 = \mathbb{R} \oplus \mathbb{R}$. Indeed, when calculating this stalk we can restrict to neighborhoods $(-\epsilon,\epsilon)$ of $0$. The pullback of such a neighborhood to $Z$ is $(\epsilon,0)\cup(0,\epsilon)$, and a section of this is determined by the values on these two subintervals (which can be anything).