  We study finite size scaling for the magnetic observables of an impurity
residing at the endpoint of an open quantum Ising chain in a transverse
magnetic field, realized by locally rescaling the magnetic field by a factor
$\mu \neq 1$. In the homogeneous chain limit at $\mu = 1$, we find the expected
finite size scaling for the longitudinal impurity magnetization, with no
specific scaling for the transverse magnetization. At variance, in the
classical impurity limit, $\mu = 0$, we recover finite scaling for the
longitudinal magnetization, while the transverse one basically does not scale.
For this case, we provide both analytic approximate expressions for the
magnetization and the susceptibility as well as numerical evidences for the
scaling behavior. At intermediate values of $\mu$, finite size scaling is
violated, and we provide a possible explanation of this result in terms of the
appearance of a second, impurity related length scale. Finally, on going along
the standard quantum-to-classical mapping between statistical models, we derive
the classical counterpart of the quantum Ising chain with an impurity at its
endpoint as a classical Ising model on a square lattice wrapped on a
half-infinite cylinder, with the links along the first circle modified as a
function of $\mu$.
