  As follows from the Schwartz Impossibility Theorem, multiplication of two
distributions is in general impossible. Nevertheless, often one needs to
multiply a distribution by a discontinuous function, not by an arbitrary
distribution. In the present paper we construct a space of distributions where
the general operation of multiplication by a discontinuous function is defined,
continuous, commutative, associative and for which the Leibniz product rule
holds. In the new space of distributions, the classical delta-function
$\delta_\tau$ extends to a family of delta-functions $\delta_\tau^\alpha$,
dependent on the \textit{shape} $\alpha$. We show that the various known
definitions of the product of the Heaviside function and the delta-function in
the classical space of distributions $\mathcal D'$ become particular cases of
the multiplication in the new space of distributions, and provide the
applications of the new space of distributions to the ordinary differential
equations which arise in optimal control theory. Also, we compare our approach
of the Schwartz distribution theory with the approach of the Colombeau
generalized functions algebra, where the general operation of multiplication of
two distributions is defined.
