 Kruzhkov entropy flux

For ${\eta }_k(z)=|z-k|$, the displayed flux is continuous at $z=k$ and has derivative $\mathrm{sgn}(z-k)f^{\prime }(z)$ almost everywhere. On a bounded state range with $|f^{\prime }|\le M$, its symmetric two-state version $Q(a,b)$ obeys $|Q(a,b)|\le M|a-b|$ and is Lipschitz in both arguments. The vanishing factor at $a=b$ removes any sign-function jump contribution.