We study the magnetic characteristics of the two dimensional (2D) square lattice with one valence electron per lattice site in weak rational magnetic fields accessible experimentally, by solving numerically the exact system of discrete equations, which fully describes the broadening of Landau levels. The 2D square lattice is used as a prototype electron system in which the Fermi level lies at the energy of the saddle point (Van Hove peak) of the Brillouin zone and the corresponding density of states N(E_F) is formally infinite (N(E_F)→+∞). According to the electron band treatment this could lead to a formally divergent paramagnetic susceptibility and an infinite electron contribution to the specific heat at zero temperature, but our accurate analysis shows that both values remain finite. The energy spectrum of the Landau level at the saddle point has turned out to be principally continuous, so that even in a very small magnetic field this Landau level is always broadened in a miniband, in contrast to other energies, where the spectrum of Landau levels is discrete. Taking into account the electron spin polarization and the numerical solution, we reproduce the temperature dependence of the induced magnetic moment proportional to the magnetic susceptibility, and the electron contribution to the specific heat. Both plots demonstrate unusual dependencies reflecting the “metal”-like or “insulator”-like structure of the Landau minibands in the neighborhood of the Fermi energy. At low temperatures all values display oscillatory behavior. We also prove rigorously that the fully occupied electron band has no contribution to the diamagnetic susceptibility and specific heat.