  We study the occurrence of spontaneous symmetry breaking (SSB) for O(N)
models using functional renormalization group techniques. We show that even the
local potential approximation (LPA) when treated exactly is sufficient to give
qualitatively correct results for systems with continuous symmetry, in
agreement with the Mermin-Wagner theorem and its extension to systems with
fractional dimensions. For general N (including the Ising model N=1) we study
the solutions of the LPA equations for various truncations around the zero
field using a finite number of terms (and different regulators), showing that
SSB always occurs even where it should not. The SSB is signalled by
Wilson-Fisher fixed points which for any truncation are shown to stay on the
line defined by vanishing mass beta functions.
