  We establish analogues of Liouville's theorem in the complex function theory,
with the differential operator replaced by various difference operators. This
is done generally by the extraction of (formal) Taylor coefficients using a
residue map which measures the obstruction having local "anti-derivative". The
residue map is based on a Weyl algebra or $q$-Weyl algebra structure satisfied
by each corresponding operator. This explains the different senses of
"boundedness" required by the respective analogues of Liouville's theorem in
this article.
