Inverse continuity on the boundary of the numerical range

Let $A\in M_n(\text{C})$. We consider the mapping $f_A(x)=x^{\ast }Ax$, defined on the unit sphere in ${\text{C}}^n$. The map has a multi-valued inverse $f_A^{-1}$, and the continuity properties of $f_A^{-1}$ 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 $f_A^{-1}$, and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for $n=4$ and a sufficient condition for $n>4$.