Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. How about $\mathscr{F}'$ if the other two sheaves are flasque? Can we prove $\mathscr{F}'$ flasque or end the question by a counterexample? Thanks for any tips.



A question on flasque sheaf
For a naturally-arising example, take $X$ to be a smooth variety and consider


$0 \to \mathcal{O}\_X^\times \to \mathcal{R}\_X^\times \to \mathcal{D}\textit{iv}\_X \to 0$


where $\mathcal{O}\_X$ is the structure sheaf, $\mathcal{R}\_X$ is the sheaf of rational functions, and $\mathcal{D}\textit{iv}\_X$ is (by definition) the sheaf of Cartier divisors.


