Propagation of a derivative jump along a characteristic ($u=A+g(q),Dq=0$)

Suppose $D=a{\partial }_x+b{\partial }_y$, $DA=f$, $Dq=0$, and $A,q$ are smooth functions. Then $u=A+g(q)$ solves $Du=f$ wherever $g$ is differentiable. If $g$ is continuous and piecewise continuously differentiable with a jump in $g^{\prime }$ at $q=q_0$, the one-sided gradients of $u$ differ by $[g^{\prime }]\nabla q$ across that characteristic curve. Thus a nonzero derivative jump in the data persists along the characteristic unless the relevant component of $\nabla q$ vanishes. A locally Lipschitz continuous $g$ also gives a weak solution across the curve.