The Stacks project

Exercise 111.36.4. Find an example of a scheme $X$, points $x, x' \in X$, a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ such that (a) $x$ is a specialization of $x'$ and (b) $\mathcal{F}_{x'} \not= 0$ and $\mathcal{F}_ x = 0$.