Using the proper definition(1) of the massless 2D-propagator - dependi
ng on an arbitrary ir-scale mu - We check Elitzur's conjecture on the
ir finiteness of O(N)-invariant Green functions. Explicit computations
to the second order in the coupling constant show that noninvariant c
orrelators become finite but mu-dependent objects, whereas the correla
tors of invariant operators, which are possible candidates for ''physi
cal'' observables, do not depend on the ir-scale.