  The gauge-fixing parameter $\xi$ dependence of two-point gauge variant
correlation functions is studied for QED and QCD. We show that, in three
Euclidean dimensions, or for four-dimensional thermal gauge theories, the usual
procedure of getting a general covariant gauge-fixing term by averaging over a
class of covariant gauge-fixing conditions leads to a nontrivial gauge-fixing
parameter dependence in gauge variant two-point correlation functions (e.g.
fermion propagators). This nontrivial gauge-fixing parameter dependence
modifies the large distance behavior of the two-point correlation functions by
introducing additional exponentially decaying factors. These factors are the
origin of the gauge dependence encountered in some perturbative evaluations of
the damping rates and the static chromoelectric screening length in a general
covariant gauge. To avoid this modification of the long distance behavior
introduced by performing the average over a class of covariant gauge-fixing
conditions, one can either choose a vanishing gauge-fixing parameter or apply
an unphysical infrared cutoff.
