  In estimating an unknown parameter of a quantum state the quantum Fisher
information (QFI) is a pivotal quantity, which depends on the state and its
derivate with respect to the unknown parameter. We prove the continuity
property for the QFI in the sense that two close states with close first
derivatives have close QFIs. This property is completely general and
irrespective of dynamics or how states acquire their parameter dependence and
also the form of parameter dependence---indeed this continuity is basically a
feature of the classical Fisher information that in the case of the QFI
naturally carries over from the manifold of probability distributions onto the
manifold of density matrices. We demonstrate that in the special case where the
dependence of the states on the unknown parameter comes from one dynamical map
(quantum channel), the continuity holds in its reduced form with respect to the
initial states. In addition, we show that when one initial state evolves
through two different quantum channels, the continuity relation applies in its
general form. A situation in which such scenario can occur is an open-system
metrology where one of the maps represents the ideal dynamics whereas the other
map represents the real (noisy) dynamics. In the making of our main result, we
also introduce a regularized representation for the symmetric logarithmic
derivative which works for general states even with incomplete rank, and its
features continuity similarly to the QFI.
