  For a large class of isotrivial rational elliptic surfaces (with section), we
show that the set of rational points is dense for the Zariski topology, by
carefully studying variations of root numbers among the fibers of these
surfaces. We also prove that these surfaces satisfy a variant of weak-weak
approximation. Our results are conditional on the finiteness of
Tate-Shafarevich groups for elliptic curves over the field of rational numbers.
