  The Luttinger Theorem, which relates the electron density to the volume of
the Fermi surface in an itinerant electron system, is taken to be one of the
essential features of a Fermi liquid. The microscopic derivation of this result
depends on the vanishing of a certain integral, the Luttinger integral $I_{\rm
L}$, which is also the basis of the Friedel sum rule for impurity models,
relating the impurity occupation number to the scattering phase shift of the
conduction electrons. It is known that non-zero values of $I_{\rm L}$ with
$I_{\rm L}=\pm\pi/2$, occur in impurity models in phases with non-analytic low
energy scattering, classified as singular Fermi liquids. Here we show the same
values, $I_{\rm L}=\pm\pi/2$, occur in an impurity model in phases with regular
low energy Fermi liquid behavior. Consequently the Luttinger integral can be
taken to characterize these phases, and the quantum critical points separating
them interpreted as topological.
