  In this paper, we establish a link between the Hyers-Ulam stability and the
Moore--Penrose inverse, that is, a closed operator has the Hyers-Ulam stability
if and only if it has a bounded Moore-Penrose inverse. Meanwhile, the stability
constant can be determined in terms of the Moore-Penrose inverse. Based on this
result, some conditions for the perturbed operators having the Hyers--Ulam
stability are obtained and the Hyers-Ulam stability constant is expressed
explicitly in the case of closed operators. In the case of the bounded linear
operators we obtain some characterizations for the Hyers-Ulam stability
constants to be continuous. As an application, we give a characterization for
the Hyers-Ulam stability constants of the semi-Fredholm operators to be
continuous.
