  We study renormalized quenched strong-coupling QED in four dimensions in
arbitrary covariant gauge. Above the critical coupling leading to dynamical
chiral symmetry breaking, we show that there is no finite chiral limit. This
behaviour is found to be independent of the detailed choice of photon-fermion
proper vertex in the Dyson-Schwinger equation formalism, provided that the
vertex is consistent with the Ward-Takahashi identity and multiplicative
renormalizability. We show that the finite solutions previously reported lie in
an unphysical regime of the theory with multiple solutions and ultraviolet
oscillations in the mass functions. This study supports the assertion that in
four dimensions strong coupling QED does not have a continuum limit in the
conventional sense.
