= Fold of a solution branch
{title2=$F=0,\quad F_x=0,\quad F_\lambda\ne0,\quad F_{xx}\ne0$}
= Continuation fold
{synonym}
A nondegenerate <fold of a solution branch> in a smooth scalar equation satisfies the displayed conditions. The <implicit function theorem> still gives $\lambda=\lambda(x)$ because $F_\lambda\ne0$. <Implicit differentiation> gives $\lambda'(x_*)=0$ and $\lambda''(x_*)=-F_{xx}/F_\lambda\ne0$: the parameter has a genuine local extremum on the <curve>.
In the cubic <polynomial>, $F_x=0$ gives $x_*=\pm1/\sqrt3$, and substituting into $F=0$ gives $\lambda_*=2x_*/3$. Furthermore, $F_{xx}=6x_*\ne0$. Yet $\nabla F=(3x^2-1,1)$ never vanishes, so the full <solution branch> remains a <regular level set> everywhere. The obstruction belongs to the projection onto $\lambda$, not to the <curve> itself.
For systems, the cited continuation account uses a one-dimensional <kernel> of $F_u$ and $F_\lambda$ outside its <image of a linear map> to define a simple fold (its Definition 2.2). These conditions make $[F_u\ F_\lambda]$ have full <rank> and support a transverse correction. A nonzero second-order term along the <kernel> is additionally needed to describe a nondegenerate quadratic turning point; in the scalar example this is $F_{xx}\ne0$.<Condition Estimates for Pseudo-Arclength Continuation by K. I. Dickson and others>[§2.2, Definition 2.2, printed p. 265: one-dimensional kernel of the state Jacobian and parameter derivative outside its range]{ref}
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