  We investigate the Exact Renormalization Group (ERG) description of ($Z_2$
invariant) one-component scalar field theory, in the approximation in which all
momentum dependence is discarded in the effective vertices. In this context we
show how one can perform a systematic search for non-perturbative continuum
limits without making any assumption about the form of the lagrangian.
Concentrating on the non-perturbative three dimensional Wilson fixed point, we
then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding
about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and
higher, yields solutions that at first converge to the answer obtained without
truncation, but then cease to further converge beyond a certain point. No
completely reliable method exists to reject the many spurious solutions that
are also found. These properties are explained in terms of the analytic
behaviour of the untruncated solutions -- which we describe in some detail.
