  By considering families of radial null geodesics, we study the subsets of
initial data that lead to naked singularities and black holes in inhomogeneous
spherical dust collapse. We introduce the notion of central homogeneity for
spherical dust collapse and prove that for the occurrence of naked
singularities, the initial data set must in general be centrally homogeneous.
Even though mathematically this indicates that naked singularities are in
general unstable, we show that centrally inhomogeneous perturbations in the
initial data are not physically reasonable. This provides an example of the
fact that instability in this context deduced with respect to general
perturbations can become stabilised once the class of perturbations are
restricted to be physical.
