  Generalized smooth functions are a possible formalization of the original
historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin,
Heaviside, and Dirac to deal with generalized functions. They are
set-theoretical functions defined on a natural non-Archimedean ring, and
include Colombeau generalized functions (and hence also Schwartz distributions)
as a particular case. One of their key property is the closure with respect to
composition. We review the theory of generalized smooth functions and prove
both the local and some global inverse function theorems.
