  Keller's theorem relates the components of the macroscopic dielectric
response of a binary two-dimensional composite system with those of the
reciprocal system obtained by interchanging its components. We present a
derivation of the theorem that, unlike previous ones, does not employ the
common asumption that the response function relates an irrotational to a
solenoidal field and that is valid for dispersive and dissipative anisotropic
systems. We show that the usual statement of Keller's theorem in terms of the
conductivity is strictly valid only at zero frequency. We verify the theorem
numerically in several ordered and disordered systems and discuss some of its
consequences.
