I tried to prove the following assertion, which I think is implicit in a text I have read, but I'm not sure about that:
Let $X$ be an integral scheme and $\mathcal F$ a locally free sheaf of finite rank on $X$.
Let $s$ be a section of $\mathcal F(X)$ which is zero in $\mathcal F(U)$, where $U$ is an open nonempty subset of $X$.
Is it true that then also $s$ itself is zero in $\mathcal F(X)$?