Inverse continuity on the boundary of the numerical range

Let (Formula presented.). We consider the mapping (Formula presented.) , defined on the unit sphere in (Formula presented.). The map has a multi-valued inverse (Formula presented.) , and the continuity properties of (Formula presented.) are considered in terms of the structure of the set of pre-images for points in the numerical range. It is shown that there may be only finitely many failures of continuity of (Formula presented.) , and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for (Formula presented.) and a sufficient condition for (Formula presented.).