  A density functional theory is developed for fermions in one dimension,
interacting via a delta-function. Such systems provide a natural testing ground
for questions of principle, as the local density approximation should work well
for short-ranged interactions. The exact-exchange contribution to the total
energy is a local functional of the density. A local density approximation for
correlation is obtained using perturbation theory and Bethe-Ansatz results for
the one-dimensional contact-interacting uniform Fermi gas. The ground-state
energies are calculated for two finite systems, the analogs of Helium and of
Hooke's atom. The local approximation is shown to be excellent, as expected.
