Limit of vanishing regulator in the functional renormalization group

The nonperturbative functional renormalization group equation depends on the choice of a regulator function, whose main properties are ``coarse-graining scale'' $k$ and an overall dimensionless amplitude $a$. In this paper we shall discuss limit $a\ensuremath{\rightarrow}0$ with fixed. This is closely related to pseudoregulator that reproduces beta functions $\overline{\mathrm{MS}}$ scheme studied in previous paper. It not suitable for precision calculations but it appears be useful eliminate spurious breaking symmetries by regulator, both nonlinear models within background field method.