  When can we uniquely map the dynamic evolution of a classical density to a
time-dependent potential? In equilibrium, without time dependence, the one-body
density uniquely specifies the external potential that is applied to the
system. This mapping from a density to the potential is the cornerstone of
classical density functional theory (DFT). Here, we derive rigorous and
explicit conditions for such a unique mapping between a nonequilibrium density
profile and a time-dependent external potential. We thus prove the underlying
assertion of dynamical density functional theory (DDFT) - with or without the
so-called adiabatic approximation often used in applications. We also
illustrate loopholes when our conditions are violated so that two distinct
external potentials result in the same density profiles but different currents,
as suggested by the framework of power functional theory (PFT).
