  We investigate when Taylor expansions can be used to prove the Runge-Gross
Theorem, which is at the foundation of Time-Dependent Density Functional Theory
(TDDFT). We start with a general analysis of the conditions for the Runge-Gross
argument, especially the time-differentiability of the density. The latter
should be questioned in the presence of singular (e.g. Coulomb) potentials.
Then, we show that a singular potential in a one-body operator considerably
decreases the class of time-dependent external potentials to which the original
argument can be applied. A two-body singularity has an even stronger impact and
an external potential is essentially incompatible with it. For the Coulomb
interaction and all reasonable initial many-body states, the Taylor expansion
only exists to a finite order, except for constant external potentials.
Therefore, high-order Taylor expansions are not the right tool to study atoms
and molecules in TDDFT.
