So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y$, a continuous map $f:X\rightarrow Y$ and a sheaf of abelian groups $\mathcal{F}$ on $X$ such that the presheaf $f_*^{i}(\mathcal{F})$ given by $$ U\mapsto H^i(f^{-1}(U),\mathcal{F}|_{f^{-1}(U)}) $$ (for some $i>0$) fails to be a sheaf on $Y$.

I'd like to see the two possible obstructions for $f_*^{i}(\mathcal{F})$ to be a sheaf namely

(i) the impossibility of gluing local sections of $f_*^{i}(\mathcal{F})$ to a global one,

(ii) the existence of a **non-zero** global section of $f_*^{i}(\mathcal{F})$ which is trivial at each stalk $y\in Y$.