We analyze a flux-limited sample of persistent and bright (with 2-10 keV fluxes exceeding 1.4e-10 erg/s/cm2) low-mass X-ray binaries (LMXBs) in our Galaxy. It is demonstrated that the majority of binary systems with X-ray luminosities below logL(erg/sec)~37.3 have unevolved secondary companions (except for those with white dwarf donors), while systems with higher X-ray luminosity predominantly harbor giant donors. Mass transfer in binary systems with giants significantly shortens their life time thus steepening the X-ray luminosity function of LMXBs at high luminosity. We argue that this is the reason why the LMXB luminosity function constructed in the last years from observations of sources in our and distant galaxies demonstrates a break at logL(erg/sec)~37.3.