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For Noncharacteristic Cauchy data parameterized by $t\ge 0$, the method of characteristics produces a surface $\Phi (s,t)=(x,y,z)$. The part uniquely fixed as a classical graph through the data must retain both the allowed footpoint range and a nonsingular branch of the projected map. On a branch where inversion gives $(s(x,y),t(x,y))$, require $t(x,y)\ge 0$, together with the branch condition and a nonzero Jacobian determinant. The strict inequality $t>0$ describes its open interior; the endpoint characteristic $t=0$ supplies a boundary trace. A fold caustic can supply another boundary.

An algebraic formula obtained by eliminating $s,t$ can extend beyond this domain of dependence. Its values there need not be selected by the original data, because their characteristic footpoints can lie on the unprescribed half of the data curve. Tracking the parameter range is therefore part of the uniqueness argument, separately from selecting a real square-root branch.

Differentiating the projected parameterization givesIt vanishes exactly when $s=-1$. The image is the tangency half-line from part (iii), $x=t-1$, $y=-t$, $t\ge 0$. This is a fold caustic: the parameterized surface itself remains regular, but its projection ceases to be locally invertible, so it cannot there be expressed as a differentiable single-valued graph over the plane.

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 $t\ge 0$. Inverting the positive branch givesThe square root must be positive, and $t\ge 0$ requires $x+1\ge \sqrt{1+x+y}>0$. Squaring is now legitimate and gives $y\le x^2+x$. Thus the full characteristic-covered set on this branch isIts open interior, where both the data parameter and the projection are interior and nonsingular, is the classical domain of dependence

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 $W\downarrow 0$.

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 $1+\epsilon h(t)$, 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.