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, however, 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 Schrödinger 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.