  We study renormalized quenched strong-coupling QED in four dimensions in
arbitrary covariant gauge, in the Dyson-Schwinger equation formalism. Above the
chiral critical coupling, we show that there is no finite chiral limit. This
behaviour is found to be independent of the detailed choice of proper vertex,
provided that the vertex is consistent with the Ward-Takahashi identity and
multiplicative renormalizability. The finite solutions previously reported lie
in an unphysical regime of the theory with multiple solutions and oscillating
mass functions. This study is consistent with the assertion that strong
coupling QED$_4$ does not have a continuum limit in the conventional sense.
