  A particularly easy, even if for long overlooked way is presented for
defining globally arbitrary Lie group actions on smooth functions on Euclidean
domains. This way is based on the appropriate use of the usual parametric
representation of functions. As a further benefit of this way, one can define
large classes of genuine Lie semigroup actions. Here "genuine" means that,
unlike in the literature, such Lie semigroups need no longer be sub-semigroups
of Lie groups, and instead, can contain arbitrary noninvertible smooth
functions on Euclidean domains.
