Differential algebraic equations (DAEs) of the form Ex = Ax + f are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the coefficient matrices E and A are allowed, in particular, this encompasses the case where the coefficient matrices are time-varying but not continuous. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem always has a unique solution, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients).