  For a complex $n$-by-$n$ matrix $A$, the numerical range $F(A)$ is the range
of the map $f_A(x) = x^*A x$ acting on the unit sphere in $\C^n$. We ask
whether the multivalued inverse numerical range map $f_A^{-1}$ has a continuous
single-valued selection defined on all or part of $F(A)$. We show that for a
large class of matrices, $f_A^{-1}$ does have a continuous selection on $F(A)$.
For other matrices, $f_A^{-1}$ has a continuous selection defined everywhere on
$F(A)$ except in the vicinity of a finite number of exceptional points on the
boundary of $F(A)$.
