points above it: $$C = \{(t, x): x \ge - \frac{t^2}{4}\}\,.$$ The parabola is the envelope of the straight lines or, equivalently, : \[line\] x(t) = c\^2 + ct. Through every point $(t_0, x_0)$ in the interior of $C$ pass two straight lines (solutions) corresponding to $c_+(t_0, x_0)$ and $c_-(t_0, x_0)$. It is useful to notice that one of the lines touches the parabola in the past (at some moment $t = {\mathfrak t}_p(t_0, x_0) < t_0$) and the other touches the parabola in the future (at some moment $t = {\mathfrak t}_f(t_0, x_0) > t_0$). Let us denote these lines (solutions) ${\mathfrak x}_p(t; t_0, x_0)$ and ${\mathfrak x}_f(t; t_0, x_0)$, respectively. They are defined for all $t\in{{\mathbb{R}}}$. The integral funnel $S(t_0, x_0)$ contains the rays ${\mathfrak x}_p(t; t_0, x_0)$ and ${\mathfrak x}_f(t; t_0, x_0)$ for $t\ge t_0$. In addition, it contains infinitely many solutions that branch off ${\mathfrak x}_f(t; t_0, x_0)$: once the line touches the parabola at $t = {\mathfrak t}_f(t_0, x_0)$, the trajectory may continue along the parabola forever, or, at any time $r > {\mathfrak t}_f(t_0, x_0)$ it may take off along the tangent line.

To every point $(t_0, x_0)$ on the boundary of $C$ (on 