The visibility function revisited

The visibility function of a compact setS E ^d assigns to eachx S the Lebesgue outer measure of its star inS. This function was introduced by G. Beer in 1972. In 1991, A. Forte Cunto characterized the points of discontinuity of the visibility function in the boundary of a planar Jordan domain. The basic intention of this paper is to extend this characterization to a compact subset ofE ^d . Under certain assumptions, it is proved here that the visibility function of such a set is continuous at a point if and only if the set of restricted visibility of this point has null Lebesgue outer measure.This paper was written as part of Argentine Research Project 01/TX38.