Solution branch

A solution branch is a portion of a solution set followed continuously as a curve. For $F(u,\lambda )=0$ with $u\in {\mathbb{R}}^n$, a full Jacobian matrix $[F_u{F}_{\lambda }]$ of rank $n$ makes the nearby solution set a one-dimensional regular level set. It need not be a graph of a function $u(\lambda )$: a continuation fold can obstruct that particular choice of parameter while leaving the solution branch regular.