Differential properties of a generalized solution to a hyperbolic system of first-order differential equations

We study some questions of the qualitative theory of solutions to differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations. The right-hand sides of these equations contain discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. It is in particular established that its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts a common approach of investigation which uses the reduction of a system of two first-order equations to a single second-order equation.