  We prove the well-posedness of the Cauchy problem for the linear differential
system of the form $x^{\prime}-A(t)x=f$, where $f$ is a distribution and $A$
possesses at most first-kind discontinuities together with all its derivatives
defined almost everywhere. The left-hand side of this system contains the
product of a distribution and, in general, a discontinuous function, which is
undefined in the classical space of the distributions with the smooth test
functions $\mathcal D'$, so the Cauchy problem has no solution in $\mathcal
D'$. In what follows, we cosider this system in the space of distributions with
the discontinuous test functions, whose elements admit continuous and
associative multiplication by functions possessing at most first-kind
discontinuities (together with all their derivatives defined almost
everywhere), and show that there exists the unique solution of the Cauchy
problem which depends continuously on $f$.
