  We discuss the use of derivative expansion techniques for the construction of
thermal effective potentials. We present a theory for which the thermal bubble
is analytic at the origin of the momentum-frequency space, although the
internal propagators in the loop have the same mass. This means that, for this
theory, the thermal effective potential is uniquely defined. We then examine a
slightly different theory for which the thermal bubble displays the usual
non-analyticity at the origin and the thermal effective potential is not
uniquely defined. For this latter theory we compare our results with those of
other works in the literature which employ the derivative expansion but find a
uniquely defined thermal effective potential. We raise several questions
concerning the interchange of the order of the perturbative and the derivative
expansions, the thermal generalization of some non-perturbative zero
temperature methods and the use of the periodicity of the external bosonic
field. Finally, we re-examine the physical interpretation given to the
imaginary part of the thermal bubble in the literature.
