= Propagation of a derivative jump along a characteristic
{title2=$u=A+g(q),\quad Dq=0$}
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 $g'$ at $q=q_0$, the one-sided <gradients> of $u$ differ by $[g']\nabla q$ 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.
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