Cauchy problem for a differential equation with piecewise smooth characteristics

The Cauchy problem is considered for an equation with first order partial derivatives and two independent variables. One of the coefficients multiplied by a partial derivative is assumed to be discontinuous. Therefore characteristics are proved to be piecewise smooth lines and hence the generalized solution of the Cauchy problem have specific properties. In particular, it is discontinuous in a certain domain and indefinite in another domain. Importance of such investigations is connected with possible applications of results in theory of probing.