  Recently the creation of novel topological states of matter by a periodic
driving field has attracted great attention. To motivate further experimental
and theoretical studies, we investigate interesting aspects of Floquet bands
and topological phase transitions in a continuously driven Harper model. In
such a continuously driven system with an odd number of Floquet bands, the
bands are found to have nonzero Chern numbers in general and topological phase
transitions take place as we tune various system parameters, such as the
amplitude or the period of the driving field. The nontrivial Floquet band
topology results in a quantized transport of Wannier states in the lattice
space. For certain parameter choices, very flat yet topologically nontrivial
Floquet bands may also emerge, a feature that is potentially useful for the
simulation of physics of strongly correlated systems. Some cases with an even
number of Floquet bands may also have intriguing Dirac cones in the spectrum.
Under open boundary conditions, anomalous counter-propagating chiral edge modes
and degenerate zero modes are also found as the system parameters are tuned.
These results should be of experimental interest because a continuously driven
system is easier to realize than a periodically kicked system.
