  Recent interest in brane world models motivates the investigation of generic
first order dilaton gravity actions, with potentials having some
non-smoothness. We consider two different types of \delta-like contributions in
the action and analyse their effects on the solutions. Furthermore a second
source of non-smoothness arises due to the remaining ambiguities in the
solutions in the separated smooth patches, after fixing all other constants by
matching and asymptotic conditions. If moreover staticity is assumed, we
explicitly construct exact solutions.
  With the methods described, for example models with point like sources or
brane world models (where the second source of non-smoothness becomes crucial),
can now be treated as non-smooth dilaton gravity theories.
