  The non-perturbative functional renormalization group equation depends on the
choice of a regulator function, whose main properties are a "coarse-graining
scale" $k$ and an overall dimensionless amplitude $a$. In this paper we shall
discuss the limit $a\to0$ with $k$ fixed. This limit is closely related to the
pseudo-regulator that reproduces the beta functions of the
$\overline{\text{MS}}\,$ scheme, that we studied in a previous paper. It is not
suitable for precision calculations but it appears to be useful to eliminate
the spurious breaking of symmetries by the regulator, both for nonlinear models
and within the background field method.
