  We show that ultraviolet divergences found in fermionic Green's functions of
massless $QED_2$ have an essentially non-perturbative nature. We investigate
their origin both in gauge invariant formalism (the one where we introduce
Wess-Zumino fields to restore quantum gauge invariance) and in gauge
non-invariant formalism, mapping two different but equivalent mechanisms
responsible for their appearance. We find the same results in both approaches,
what contradicts a previous work of Jian-Ge, Qing-Hai and Yao-Yang, that found
no divergences in the chiral Schwinger model considered in the gauge invariant
formalism.
