Point assignments do not alter weak jump

Assume the Axiom of Countable Choice. The assertion that a pointwise value assigned on a shock plane determines its one-sided trace or weak jump is false. Let $d,m\ge 1$, let $I\subseteq \mathbb{R}$ and $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, let $\mathcal{O}\subseteq {\mathbb{R}}^m$ be open and convex, and let $F_j\in C^1(\mathcal{O};{\mathbb{R}}^m)$ for $1\le j\le d$. Let $n\in {\mathbb{R}}^d$ be a unit vector, let $s\in \mathbb{R}$, put $Q=I\times \mathrm{\Omega }$, and set $\mathrm{\Sigma }=\{(t,x)\in Q:n\cdot x=st\}$. Suppose $U:Q\to \mathcal{O}$ is measurable, has piecewise $C^1$ side fields with $C^1$ extensions to the interface, and $U,F_1(U),\dots ,F_d(U)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(Q)$. Let $\tilde{U}:Q\to \mathcal{O}$ be any measurable field that agrees with $U$ off $\mathrm{\Sigma }$. Then $\tilde{U}$ and $F_j(\tilde{U})$ are locally integrable and define the same regular distributions as $U$ and $F_j(U)$; their distributional conservation residuals are equal. For every convex $\eta \in C^2(\mathcal{O})$ and compatible $q_j\in C^1(\mathcal{O})$ with $D{q}_j=D\eta D{F}_j$ whose compositions $\eta (U),q_j(U)$ are locally integrable, the corresponding compositions for $\tilde{U}$ are locally integrable and have the same regular distributions and entropy residual. The two fields have the same one-sided traces and jump. In particular, assigning a third state on the shock plane does not change its weak jump condition.

piecewise smooth traces and jump convention

regular distributions and weak formulations

Lebesgue nullity of an affine hyperplane

Axiom of Countable Choice