= Solution
Characteristics beginning at $\xi$ have $x=\xi+t u_I(\xi)$ and carry the constant value $u_I(\xi)$. In the sloping interval, $x=t+(1-t)\xi$. For $0\leq t<1$ the resulting single-valued solution is
$$
\boxed{u(x,t)=\begin{cases}1,&x\leq t,\\(1-x)/(1-t),&t<x<1,\\0,&x\geq1.\end{cases}}
$$
It is continuous and piecewise smooth, locally Lipschitz for $t<1$, but not a classical $C^1$ solution across the two <derivative> corners. Its weak <derivatives> satisfy the equation, since the function itself has no jumps there. The sloping gradient is $-1/(1-t)$ and blows up at $t=1$, when those characteristics coalesce.
Thus \b[the continuous characteristic description has maximal interval $0\leq t<1$]. If “<weak solution>” is taken literally, it does not cease to exist then: the <entropy> continuation is a shock with left/right states 1 and 0 and Rankine-Hugoniot speed $1/2$, situated at $x=(t+1)/2$ for $t\geq1$. It is a global <entropy> <weak solution>, with the step at $t=1$ as its limiting trace. This distinguishes breakdown of the characteristic parametrization from nonexistence of a weak continuation.
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