  For a function $F: X \to Y$ between real Banach spaces, we show how
continuation methods to solve $F(u) = g$ may improve from basic understanding
of the critical set $C$ of $F$. The algorithm aims at special points with a
large number of preimages, which in turn may be used as initial conditions for
standard continuation methods applied to the solution of the desired equation.
A geometric model based on the sets $C$ and $F^{-1}(F(C))$ substantiate our
choice of curves $c \in X$ with abundant intersections with $C$.
  We consider three classes of examples. First we handle functions $F: R^2 \to
R^2$, for which the reasoning behind the techniques is visualizable. The second
set of examples, between spaces of dimension 15, is obtained by discretizing a
nonlinear Sturm-Liouville problem for which special points admit a high number
of solutions. Finally, we handle a semilinear elliptic operator, by computing
the six solutions of an equation of the form $-\Delta - f(u) = g$ studied by
Solimini.
