  We analyze $(2+1)$-dimensional vector-vector type four-Fermi interaction
(Thirring) model in the framework of the $1/N$ expansion. By solving the
Dyson-Schwinger equation in the large-$N$ limit, we show that in the
two-component formalism the fermions acquire parity-violating mass dynamically
in the range of the dimensionless coupling $\alpha$, $0 \leq \alpha \leq
\alpha_c \equiv {1\over16} {\rm exp} (- {N \pi^2 \over 16})$. The symmetry
breaking pattern is, however, in a way to conserve the overall parity of the
theory such that the Chern-Simons term is not induced at any orders in $1/N$.
$\alpha_c$ turns out to be a non-perturbative UV-fixed point in $1/N$. The
$\beta$ function is calculated to be $\beta (\alpha) = -2 (\alpha - \alpha_c)$
near the fixed point, and the UV-fixed point and the $\beta$ function are shown
exact in the $1/N$ expansion.
