The notation is from O. Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981.
Let $\mathcal{F}$ be a presheaf on the topological space $X$. Let $U\subset X$. Let $s:U\rightarrow |\mathcal{F}|$ be a section of $p$ over $U$. Show that $s$ is continuous if and only if for every $x\in U$, there exists an open neighbourhood $V$ of $x$ and $f\in\mathcal{F}(V)$ such that $s(v)=\rho_v(f)$ for all $y\in V$.
I have attempted both direction with no success. In the forward direction, I can't seem to find an opening to use the condition of $s$ being continuous. On the backwards direction, all I got is that I need to find $s^{-1}([U,f])$. So any kind of assistance would be great.
I'll do one direction, and that should give you the idea of how to approach the other direction.
Suppose that $s: U \to |\mathcal{F}|$ is continuous. This means that $s^{-1}$ of an open set is open. Taken a basic open set $[V,f]$ around $\rho_{x}(f)$ in $s(U)$. Then consider the open set $W = s^{-1}(V)$. This contains $x$ by construction, and is open by the continuity of $s$. Moreover, for all $w \in W$ we have $s(w) = \rho_{w}(f)$ by the definition of the open set $[V,f]$.