= Solution
{c}
Every point with $x>0$ has exactly one backward <characteristic curve> meeting $x=1$, at the point $(1,q(x,y))$. It meets the specified data precisely when $-1\leq q\leq1$. Thus the whole region determined by these data, including the two bounding <characteristic curves>, is
$$
\boxed{\mathcal D_\beta=\{(x,y):x>0,\ (\beta-1)x^2-\beta x\leq y\leq(\beta+1)x^2-\beta x\}.}
$$
Its interior is the corresponding open domain for the <partial differential equation>. Uniqueness here means uniqueness in the class of <continuous functions> satisfying the <characteristic equations for a transport equation>, including the piecewise <smooth function> defining the <solution of a differential equation> above. The endpoints of the initial segment determine the two boundary values by <continuity>.
This region is maximal for determination by the given data: on $x>0$, arbitrary extensions of $g$ outside $[-1,1]$ give different values on the <characteristic curves> with $q\notin[-1,1]$. No <characteristic curve> from $x=1$ reaches $x\leq0$.
For $\beta=1$,
$$
\boxed{x>0,\qquad -x\leq y\leq2x^2-x.}
$$
The lower boundary is a straight line, the upper boundary is a <parabola>, and the characteristic carrying the corner is $y=x^2-x$. All three approach the excluded origin as $x\downarrow0$.
\Image[/past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2021-characteristic-region.png]
{title=The characteristic region for beta equal to one, with the curve carrying the corner}
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