  We consider anomaly free combinations of chiral fermions coupled to $U(1)$
gauge fields on a 2D torus first in the continuum and then on the lattice in
the overlap formulation. Both in the continuum and on the lattice, when the
background consists of sufficiently large constant gauge potentials the action
induced by the fermions varies significantly under certain singular gauge
transformations. ``Ruling away'' such discontinuities cannot be justified in
the continuum framework and does not naturally fit on the lattice. Complete
gauge invariance in the continuum can be restored in some models by choosing
special boundary conditions for the fermions. Evidence is presented that gauge
averaging the overlap phases in these models produces correct continuum
results.
