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We study some analytical properties of the solutions of the non perturbativerenormalization group flow equations for a scalar field theory with $Z_2$symmetry in the ordered phase, i.e. at temperatures below the criticaltemperature. The study is made in the framework of the local potentialapproximation. We show that the required physical discontinuity of the magneticsusceptibility $\chi(M)$ at $M=\pm M_0$ ($M_0$ spontaneous magnetization) isreproduced only if the cut-off function which separates high and low energymodes satisfies to some restrictive explicit mathematical conditions; we stressthat these conditions are not satisfied by a sharp cut-off in dimensions ofspace $d<4$.

The non-perturbative renormalization group in the ordered phase