This is the problem 2.6 of Tennison Sheaf Theory.
Let $K$ be infinite field(not necessarily algebraically closed). Endow $K$ zariski topology. Define sheaf of regular functions on $K$ as for $U$ open in $K$, $O(U)=\{\frac{f}{g},f,g\in K[t],\forall p\in U,g(p)\neq 0\}$.
Consider set of continuous sections from $U$ to sheaf space formed by $O$. Show that direct limit over any non-empty open sets $U_i$ ordered by reverse inclusion $lim\Gamma(U_i,O)=K(t)$.
Why this $K(t)$? I cannot guarantee that there is no $p\in \cap U_i$. If that is the case, I cannot invert $x-p$ here. Did I miss something here?
You are supposed to be taking the direct limit over the poset of all nonempty open subsets of $K$, not "any" nonempty open subsets as you wrote. The intersection of all nonempty open subsets is empty, since for any $p\in K$, $K\setminus\{p\}$ is open.