Is the sheaf of locally constant functions flasque?

Question

Quick question, is the sheaf of locally constant functions flasque?

Answer 1

No, take two disjoint open sets $U$ and $V$ lying in the same connected component $X_0$ of the entire space $X$. Then define a section on $U \cup V$ by the function being $0$ on $U$ and 1 on $V$. Then this section cannot extend to $X$.

Answer 2

This is true if your space is irreducible. Kedlaya gives the example of $U = \mathbf{R} - \{0\}$ sitting inside of $\mathbf{R}$ with the Euclidean topology, showing that a constant sheaf need not be flasque in general.