  A cornerstone of current-density functional theory (CDFT) in its paramagnetic
formulation is proven. After a brief outline of the mathematical structure of
CDFT, the lower semi-continuity and expectation valuedness of the CDFT
constrained-search functional is proven, meaning that there is always a
minimizing density matrix in the CDFT constrained-search universal density
functional. These results place the mathematical framework of CDFT on the same
footing as that of standard DFT.
