  Precise band-filling control is performed on a strongly correlated electron
system NaxCoO2, by changing the Na content, and the influence of this on the
magnetic and thermodynamic properties is studied. It has been discovered that a
phase boundary between a Pauli paramagnetic metal appearing at small x and a
Curie-Weiss metal at large x is located within an extremely narrow range of
  0.620 < x < 0.621. The transition between these is a Lifshitz transition of
the pocket-vanishing type, exceptionally achieved by band-filling control.
Abrupt changes in various properties across the boundary are remarkably
observed, suggesting that the Lifshitz transition occurs discontinuously in
NaxCoO2, differently from the conventional Lifshitz transition defined at T =
0.
