  In this paper we present some theorems for a class of non--hyperbolic fixed
points on ${\bf R}^N$ and then analyze a family of functions $f_{\theta}$ on
the plane which have a non--hyperbolic fixed point in the origin. The dynamical
properties of the family near the fixed point, like the basin of attraction,
are studied. Finally the limits of applicability of the characterization by the
eigenvalues of the Jacobian are discussed.
