  We characterize all real matrix semigroups satisfying a mild boundedness
assumption, without assuming continuity. Besides the continuous solutions of
the semigroup functional equation, we give a description of solutions arising
from non-measurable solutions of Cauchy's functional equation. To do so, we
discuss the primary decomposition and the Jordan-Chevalley decomposition of a
matrix semigroup. Our motivation stems from a characterization of all
multi-dimensional self-similar Gaussian Markov processes, which is given in a
companion paper.
