= Solution
{c}
For $\beta=1$, $q=(y+x-x^2)/x^2$. On the two sides of the corner-carrying <characteristic curve>, the <partial derivative> is
$$
\boxed{u_y(x,y)=\begin{cases}x^{-2},&-x<y<x^2-x,\\0,&x^2-x<y<2x^2-x,\end{cases}\qquad x>0.}
$$
At every point $(x,x^2-x)$ with $x>0$, the two one-sided <partial derivatives> are $x^{-2}$ and $0$. Therefore $u_y$ does not exist on this curve; its piecewise values have a <jump discontinuity> of magnitude $x^{-2}$. There are no other interior discontinuities. The values on the outer boundaries are interpreted from within $\mathcal D_1$.
This is exactly <propagation of a derivative jump along a characteristic>. Already on $x=1$, the data have slopes $1$ and $0$ on the two sides of $y=0$. The <transport equation> transports this corner along the <characteristic curve> from $(1,0)$, which is $y=x^2-x$; it provides no smoothing. This is consistent with <Noncharacteristic Cauchy data> and does not contradict the smooth-data theorem, whose differentiability hypothesis fails here.
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