TITLE: Can every element in the stalk be represented by a section in the top space?
QUESTION [5 upvotes]: Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map?

REPLY [4 votes]: This is only true in pathological cases. Neither in complex analysis (see above) nor in algebraic geometry: If we take the structure sheaf $\mathcal{O}_{Spec(A)}$ of an affine scheme, then the question is whether the localzation maps $A \to A_{\mathfrak{p}}$ are surjective, which is almost never true.