  We study the mapping between time-dependent densities and potentials for
noninteracting electronic systems on lattices. As discovered recently by Baer
[J. Chem. Phys. 128, 044103 (2008)], there exist well-behaved time-dependent
density functions on lattices which cannot be associated with any real
time-dependent potential. This breakdown of time-dependent V-representability
can be tracked down to problems with the continuity equation which arise from
discretization of the kinetic-energy operator. Examples are given for lattices
with two points and with N points, and implications for practical numerical
applications of time-dependent density-functional theory are discussed. In the
continuum limit, time-dependent noninteracting V-representability is restored.
