I wonder if it is possible to prove the sheaf of discontinuous sections over a paracompact Hausdorff space is acyclic with respect to Cech cohomology from the definition?
i.e. given a sheaf $\mathscr{A}$ over a paracompact Hausdorff space $M$ and a locally finite open cover $\mathcal{U}=\{U_i\}_{i\in I}$, then $$ H^p(\mathcal{U},\mathcal{C}(\mathscr{A}))=0,\quad p>0 $$ where $\mathcal{C}(\mathscr{A})$ is the sheaf of discontinuous sections of $\mathscr{A}$.
Recall that a sheaf $\mathscr{S}$ is soft if any section of $\mathscr{S}$ over a closed subset $K \subset M$ can be extended to a section of $\mathscr{S}$ over all of $M$. It is clear that the sheaf of discontinuous sections is a soft sheaf. It, therefore, suffices to show that soft sheaves have trivial cohomology (i.e., are acylic). This is contained in Proposition 2, Chapter VI of Gunning and Rossi's Analytic Functions of Several Complex Variables, page 173.