  The response of a one-dimensional fermion system is investigated using
Density Functional Theory (DFT) within the Local Density Approximation (LDA),
and compared with exact results. It is shown that DFT-LDA reproduces
surprisingly well some of the characteristic features of the Luttinger liquid,
namely the vanishing spectral weight of low energy particle-hole excitations,
as well as the dispersion of the collective charge excitations. On the other
hand, the approximation fails, even qualitatively, for quantities for which
backscattering is important, i.e., those quantities which are crucial for an
accurate description of transport. In particular, the Drude weight in the
presence of a single impurity is discussed.
