  Chiral symmetry breaking in a purely fermionic theory is investigated by the
help of the renormalization group method. The RG equation for the running mass
$m_k$ admits a solution with vanishing bare mass and finite physical mass. The
running Fermi coupling constant, $G_k$, converges to a finite (renormalized)
physical value. It is also shown that the RG equation for $\tilde G$, the
dimensionless Fermi coupling, has an UV fixed point $\tilde G_{UV}$. Contrary
to a previous result however, it is proven that the chiral symmetry breaking
point $\tilde G_c$ does not coincide with $\tilde G_{UV}$.
