  For an $n$-by-$n$ matrix $A$, let $f_A$ be its "field of values generating
function" defined as $f_A\colon x\mapsto x^*Ax$. We consider two natural
versions of the continuity, which we call strong and weak, of $f_A^{-1}$ (which
is of course multi-valued) on the field of values $F(A)$. The strong continuity
holds, in particular, on the interior of $F(A)$, and at such points $z \in
\partial F(A)$ which are either corner points, belong to the relative interior
of flat portions of $\partial F(A)$, or whose preimage under $f_A$ is contained
in a one-dimensional set. Consequently, $f_A^{-1}$ is continuous in this sense
on the whole $F(A)$ for all normal, 2-by-2, and unitarily irreducible 3-by-3
matrices. Nevertheless, we show by example that the strong continuity of
$f_A^{-1}$ fails at certain points of $\partial F(A)$ for some (unitarily
reducible) 3-by-3 and (unitarily irreducible) 4-by-4 matrices. The weak
continuity, in its turn, fails for some unitarily reducible 4-by-4 and
untiarily irreducible 6-by-6 matrices.
