One of the advantages of working with a sheaf $\mathcal{F}$ instead of a presheaf is that every section $s \in \mathcal{F}(U)$ is determined by the family of its germs $\{ s_x \}_{x \in U}$. I know this may fail for a presheaf in general, since you don't have the gluing axiom. Can anyone show me an example of this phenomenon?
(Maybe I want my base space $X$ to satisfy some separation axiom, to avoid trivial examples; of course trivial examples are welcome too.)
On
Georges Elencwajg has given a correct example, but here's a simpler one that might also be helpful. Let $X$ be a discrete space with just two points $p$ and $q$. Define a presheaf $F$ by letting $F(\{p,q\})=\{0,1\}$ while $F(\{p\})=F(\{q\})=F(\varnothing)=\{0\}$; the restriction maps are the only possible ones. Then both of the stalks are singletons, so the two gobal sections have the same stalk at each point.
Consider the sheaf $\mathcal C$ of real-valued continuous functions on $\mathbb R$ and its subpresheaf $\mathcal C_b$ of bounded functions, defined by the requirement that on an open subset $U\subset \mathbb R$ the sections of $\mathcal C_b(U)$ consist in the bounded continuous functions $U\to \mathbb R$.
The quotient presheaf $\mathcal F$ defined by $U\mapsto \mathcal F(U)=\mathcal C(U)/\mathcal C_b(U) $ is an example of the type of presheaf that you want.
Indeed all $\mathcal F_x \;(x\in \mathbb R)$ are zero because any continuous function defined on a neighbourhood of $x$ is bounded in a suitable smaller neighborhood of $x$.
However any global continuous unbounded function $f\in \mathcal C (\mathbb R) $ (like $x$ or $e^x$) is an example of a non- zero class in $\mathcal F(\mathbb R)$: $$0\neq \bar f\in \mathcal F(\mathbb R)=\mathcal C(\mathbb R)/\mathcal C_b(\mathbb R) \;\text {although all } (\bar f)_x=0_x\in \mathcal F_x=\{0_x\}$$