  We show that it is possible, in opposition to a previous conjecture, to
derive a Nielsen identity for the effective action in the case of the
generalized $R_\xi $-gauge, where the gauge function explicitly depends on the
gauge parameter $\xi $. Also the Nielsen identity for the effective potential
is verified to one-loop in the Abelian Higgs model and the corresponding
identity for the physical Higgs mass is derived.
