  A recent study of dynamical chiral symmetry breaking in N-flavour QED$_3$ at
finite temperature is extended to include the effect of fermion wavefunction
renormalisation in the Schwinger-Dyson equations. The simple ``zero-frequency''
truncation previously used is found to lead to unphysical results, especially
as $T \to 0$. A modified set of equations is proposed, whose solutions behave
in a way which is qualitatively similar to the $T=0$ solutions of Pennington et
al. [5-8] who have made extensive studies of the effect of wavefunction
renormalisation in this context, and who concluded that there was no critical
$N_c$ (at T=0) above which chiral symmetry was restored. In contrast, we find
that our modified equations predict a critical $N_c$ at $T \not= 0$, and an
$N-T$ phase diagram very similar to the earlier study neglecting wavefunction
renormalisation. The reason for the difference is traced to the different
infrared behaviour of the vacuum polarisation at $T=0$ and at $T \not= 0$.
