  We study some analytical properties of the solutions of the non perturbative
renormalization group flow equations for a scalar field theory with $Z_2$
symmetry in the ordered phase, i.e. at temperatures below the critical
temperature. The study is made in the framework of the local potential
approximation. We show that the required physical discontinuity of the magnetic
susceptibility $\chi(M)$ at $M=\pm M_0$ ($M_0$ spontaneous magnetization) is
reproduced only if the cut-off function which separates high and low energy
modes satisfies to some restrictive explicit mathematical conditions; we stress
that these conditions are not satisfied by a sharp cut-off in dimensions of
space $d<4$.
