  We investigate the classical and quantum Proca field (a massive vector
potential) of mass $m>0$ in arbitrary globally hyperbolic spacetimes and in the
presence of external sources. We motivate a notion of continuity in the mass
for families of observables $\left\{O_m\right\}_{m>0}$ and we investigate the
massless limit $m\to0$. Our limiting procedure is local and covariant and it
does not require a choice of reference state. We find that the limit exists
only on a subset of observables, which automatically implements a gauge
equivalence on the massless vector potential. For topologically non-trivial
spacetimes, one may consider several inequivalent choices of gauge equivalence
and our procedure selects the one which is expected from considerations
involving the Aharonov-Bohm effect and Gauss' law. We note that the limiting
theory does not automatically reproduce Maxwell's equation, but it can be
imposed consistently when the external current is conserved. To recover the
correct Maxwell dynamics from the limiting procedure would require an
additional control on limits of states. We illustrate this only in the
classical case, where the dynamics is recovered when the Lorenz constraint
remains well behaved in the limit.
