The natural parameter fails at the fold $x^2-\lambda =0$

Solving for the parameter gives exactly $\lambda =x^2$. At $x=0$ its derivative is $d\lambda \mathrm{/}dx=0$, and its second derivative is $2$. Thus the projection from the local solution curve to the $\lambda$-axis has zero derivative at the fold.

For every $\lambda <0$ there is no real solution, while for every $\lambda >0$ there are two, $x=\sqrt{\lambda }$ and $x=-\sqrt{\lambda }$. Hence no open interval about $0$ is the parameter domain of a single-valued local solution graph covering the whole nearby solution set. On either one-sided branch, $\mathrm{\mid }dx\mathrm{/}d\lambda \mathrm{\mid }=1\mathrm{/}(2\sqrt{\lambda })$ is unbounded as $\lambda \downarrow 0$.

The fold conditions are $F_x(0,0)=0$, $F_{\lambda }(0,0)=-1\mathrm{\ne }0$, and $F_{xx}(0,0)=2\mathrm{\ne }0$, so [F1] gives the same nonzero curvature conclusion independently of the explicit graph calculation.