= Solution
To find the <characteristic domain from data on a half-line>, retain the allowed range of its initial parameter. The portion connected to the initial data before the <fold caustic> has $s>-1$ and $t\geq0$. Inverting the positive branch gives
$$
s=-1+\sqrt{1+x+y},\qquad t=x+1-\sqrt{1+x+y}.
$$
The square root must be positive, and $t\geq0$ requires $x+1\geq\sqrt{1+x+y}>0$. Squaring is now legitimate and gives $y\leq x^2+x$. Thus the full characteristic-covered set on this branch is
$$
\boxed{x>-1,\qquad -x-1<y\leq x^2+x.}
$$
Its open interior, where both the data parameter and the projection are interior and nonsingular, is the classical <domain of dependence>
$$
\boxed{D=\{(x,y):x>-1,\ -x-1<y<x^2+x\}.}
$$
\Image[/past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2012-characteristics.png]
{title=The <domain of dependence> from the initial half-line, bounded by the endpoint characteristic and the <fold caustic>}
Each point of $D$ has exactly one backward characteristic meeting the interior $t>0$ of the prescribed data, so the <Picard-Lindelöf theorem> fixes its value. The upper boundary is the characteristic issuing from the endpoint $t=0$ and is fixed by the endpoint datum, consistently with the continuous trace from $D$. The lower boundary is excluded: $s=-1$ there, the <Jacobian determinant> vanishes, and the explicit solution <derivatives> diverge as $W\downarrow0$.
Points of the half-plane $W>0$ outside this set have $t<0$ on the branch through the data. Their characteristic footpoints would lie on the unprescribed continuation of the initial line, so these values are not determined by the given half-line alone. For example, extend the initial value to negative $t$ as $1+\varepsilon h(t)$, where $h$ is a <smooth function> supported on the negative half-line. Such extensions agree with all the prescribed data and, where noncharacteristic, produce different local extensions by the <method of characteristics>. The displayed formula supplies one possible extension there, but the data do not select it uniquely. This is why retaining the restriction on the initial half-line is essential. Continuing the parameterized surface past $s=-1$ instead produces the other sheet of the fold; it does not give a differentiable graph continuation through the caustic.
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