Context
In section 5.6 of Warner's Book the relationship between sheaves (the author calls sheaves to what is usually defined as Étale spaces) and presheaves is discussed. Given a sheaf S, we define its presheaf of sections $\alpha(S)$. And given a presheaf $F$ we define its associated sheaf $\beta(F)$ as the Étale space consisting of all the germs of sections. The objective is to prove that $S$ and $\beta(\alpha(S))$ are canonically isomorphic (as Étale spaces) and the book itself gives the isomorphism: Let $\xi\in \beta(\alpha(S))$ be the germ at $x$ of some section $f$ of $S$ over an open set containing $x$, that is $\xi=f_x$, for $\Gamma(S,U)$. Then $\psi: \xi \mapsto f(x)$ is the candidate to isomorphism. The exercise 5.8 consists on proving the details.
Question
I have proved that the proposed map is well defined, a local homeomorphism and surjective. All that is left to prove that $\psi$ is injective. I have seen that if $\psi(f_x)=\psi(g_y)$ then $x=y$. But I don't know how to deduce that if two sections $f\in \Gamma(S,U)$, $g \in \Gamma(S,U')$ have the same value at $x$ then both must agree in a neighborhood of $x$. I suspect that this would be false in general if we were not talking about sections with values on an Étale, (which has discrete topology in each fiber) but I don't know how to finish the argument.