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We discuss the phenomenon of spontaneous symmetry breaking by means of a class of non-perturbative renormalization group flow equations which employ a regulating smearing function in the proper-time integration. We show, both analytically and numerically, that the convexity property of the renormalized local potential is obtained by means of the integration of arbitrarily low momenta in the flow equation. Hybrid Monte Carlo simulations are performed to compare the lattice effective potential with the numerical solution of the renormalization group flow equation. We find very good agreement both in the strong and in the weak coupling regime. [Copyright &y& Elsevier]