Let $L$ be a line bundle on a smooth curve X (reduced, irreducible, scheme over a field $k$, or whatever). Then consider the sheaf $L \otimes_{O_X} K_X$, the sheaf of rational sections of $L$. Here $K_X$ is the sheaf of rational functions, defined on each open set to be just the field of rational functions $K(X)$.
From the point of view of a rational section as a section that is just defined on an open set (+ an equivalence relation), it is obvious that there are global rational sections of this new line bundle, since $L$ is locally free. However, I don't see this from the point of view of sheaf theory.
The book I am reading insists that one obviously comes from the local trivializations. I believe it, but I do not see how this works formally sheaf theoretically - I don't see why they patch together correctly.