We consider a modulated discrete nonlinear Schr\"odinger (DNLS) model with
alternating on-site potential, having a linear spectrum with two branches
separated by a 'forbidden' gap. Nonlinear localized time-periodic solutions
with frequencies in the gap and near the gap -- discrete gap and out-gap
breathers (DGBs and DOGBs) -- are investigated. Their linear stability is
studied varying the system parameters from the continuous to the
anti-continuous limit, and different types of oscillatory and real
instabilities are revealed. It is shown, that generally DGBs in infinite
modulated DNLS chains with hard (soft) nonlinearity do not possess any
oscillatory instabilities for breather frequencies in the lower (upper) half of
the gap. Regimes of 'exchange of stability' between symmetric and antisymmetric
DGBs are observed, where an increased breather mobility is expected. The
transformation from DGBs to DOGBs when the breather frequency enters the linear
spectrum is studied, and the general bifurcation picture for DOGBs with tails
of different wave numbers is described. Close to the anti-continuous limit, the
localized linear eigenmodes and their corresponding eigenfrequencies are
calculated analytically for several gap/out-gap breather configurations,
yielding explicit proof of their linear stability or instability close to this
limit.Comment: 17 pages, 12 figures, submitted to Eur. Phys. J.