$J^*(x)$ of course is not continuous. Since on non-defect sites (which correspond to bottlenecks of size $l=0$) the transport capacity is $J^*=J^{\rm max}_{\rm hom}$, it is sufficient to check condition (\[init\_cond\_1\]) for defect sites. Their number is finite in finite systems but can be infinite in the continuum limit (e.g. for disordered systems with finite defect density).

The problem of condition (\[init\_cond\_1\]) is that it is given as an inequality and does not provide initial conditions for (\[dJdx\]) on the defect sites. We now want to show that (\[init\_cond\_1\]) is identically fullfilled by a set of initial conditions $$\label{init_cond_2}
J(x) = J^*(x) \hspace{5mm} \mbox{ at defect sites $x$} \,\, ,$$ if one assumes additionally that the physical local solution at $x$ is selected by shock dynamics.

First of all, if we assume the conditions (\[init\_cond\_2\]) we see that, in contrast to the boundary conditions of the system which are usually given by a fixed density, the initial condition imposed by a defect provides the possibility of two realizations of the local density profile. Given the initial condition $J(x_0)=J^*(x_0)$ at a point $x_0$, only the current is a fixed initial condition while, due to the non-unique inversion of the current-density relation 