Differential equations with distributions admitting multiplication on discontinuous functions

The functions possessing one-sided limits in each point of open interval is considered as the test functions. The distributions are the linear continuous functionals on the introduced space of test functions. The product of distribution and discontinuous function is well defined in this space. We also consider the initial value problem for linear differential equation with distributional coefficients from this space. The theorem of existence and uniqueness of solution of this problem have proven.