I studied the following theorem:(Rankine-Hugoniot condition)
Let $u:\mathbb{R} \times [0,+\infty) \rightarrow \mathbb{R} $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if the two of the following conditions are satisfied:
i) $u$ is a classical solution of in the domain where $u$ is $C^1$
ii) $u$ satisfies the jump condition
$$(u_+ -u_-) \eta_t +\sum\limits_{j=1}^d{f_j(u_+) -f_j(u_-)} \eta_x=0$$ My doubts...
i)Can we find an example of the conservation law where solutions are not piecewise $C^1$(i.e the solution does not have a piece wise $C^1$ representative) so that we cannot apply Rankine-Hugoniot condition across the jump?
ii) Is there any weaker versions of this theorem so that piecewise $C^1$ can be relaxed?