  We describe continuity properties of the multivalued inverse of the numerical
range map $f_A:x \mapsto \left\langle Ax, x \right\rangle$ associated with a
linear operator $A$ defined on a complex Hilbert space $\mathcal{H}$. We prove
in particular that $f_A^{-1}$ is strongly continuous at all points of the
interior of the numerical range $W(A)$. We give examples where strong and weak
continuity fail on the boundary and address special cases such as normal and
compact operators.
