Let $S$ be a discrete topological space, and let $\underline{S}$ be the constant sheaf on a topological space $X$, meaning $\underline{S}(U)$ is the collection of continuous maps $U\to S$.
Am I correct in thinking that $\underline{S}(U)\cong S^n$ where $n$ is the number of connected components of $U$? Say $U=V\cup W$ where $V\cap W=\emptyset$, then we can have $s|_V:V\to \{p\}$ and $s|_W:W\to \{q\}$ where this is clearly continuous.
If the connected components of $U$ are open sets, then this is correct, since a continuous map $U\to S$ is uniquely determined by choosing a constant value on each connected component of $U$. In particular, this is always the case if $U$ has only finitely many connected components.
However, if the connected components are not open, then not every choice of a constant value on each connected component will give a continuous map $U\to S$. For instance, if $U$ is a Cantor set, then the connected components are just singletons, but not every map $U\to S$ is continuous.