  Floquet states have been used to describe the impact of periodic driving on
lattice systems, either using a tight-binding model, or by using a continuum
model where a Kronig-Penney-like description has been used to model spatially
periodic systems in one dimension. A number of these studies have focused on
finite systems, and results from these studies are distinct from those of
infinite lattice systems as a consequence of boundary effects. In the case of a
finite system, there remains a discrepancy in the results between tight-binding
descriptions and continuous lattice models. Periodic driving by a
time-dependent field in tight-binding models results in a collapse of all
quasienergies within a band at special driving amplitudes. In the continuum
model, on the other hand, a pair of nearly-degenerate edge bands emerge and
remain gapped from the bulk bands as the field amplitude increases. We resolve
these discrepancies and explain how these edge bands represent Schr\"odinger
cat-like states with effective tunneling across the entire lattice. Moreover,
we show that these extended cat-like states become perfectly localized at the
edge sites when the external driving amplitude induces a collapse of the bulk
bands.
