  The paper deals with scalar conservation laws having a flux discontinuity at
$x=0$ without a weak solution that satisfies the classical Rankine--Hugoniot
jump condition at $x=0$. We are using unbounded solutions in the form of shadow
waves supported by the origin for solving that problem. The shadow waves are
nets of piecewise constant functions approximating a shock wave with added a
delta function and sometimes another unbounded part.
