= Solution
{c}
Parametrize the initial curve by $(x_0(r),y_0(r))=(1,r)$, $-1\leq r\leq1$. Its tangent is $(0,1)$, whereas the projected <characteristic vector field> there is $(1,2r+\beta)$. The <determinant>
$$
\boxed{\det\begin{pmatrix}1&0\\2r+\beta&1\end{pmatrix}=1\ne0}
$$
proves that these are <Noncharacteristic Cauchy data>: the initial curve is everywhere transverse to the <characteristic curves>.
There is a regularity qualification. The prescribed <function> $g(r)=\min\{r,0\}$ is <continuous> and <Lipschitz continuous>, but is not differentiable at $r=0$. Thus the data are Cauchy in this geometric sense, with data given by a <smooth function> on each of $r<0$ and $r>0$; the usual theorem giving a global <classical solution> from continuously differentiable data does not apply across $r=0$.
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