  Let $A \in M_n(\C)$. We consider the mapping $f_A(x)=x^*Ax$, defined on the
unit sphere in $\C^n$. 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 $n>4$.
