  On the lattice some of the salient features of pure gauge theories and of
gauge theories with fermions in complex representations of the gauge group seem
to be lost. These features can be recovered by considering part of the theory
in the continuum. The prerequisite for that is the construction of continuum
gauge fields from lattice gauge fields. Such a construction, which is gauge
covariant and complies with geometrical constructions of the topological charge
on the lattice, is given in this paper. The procedure is explicitly carried out
in the $U(1)$ theory in two dimensions, where it leads to simple results.
