  The consequences of the constraints which de Sitter embedding of $f(R)$
theories imposes on the Lagrangian's parameters, are investigated within the
metric formalism. It is shown, in particular, that several common $f(R)$
Lagrangians do not actually admit such an embedding. Otherwise, asymptotic
matching of local solutions of the corresponding models with background
(maximally symmetric) spaces of constant curvature is either unstable or,
anti-de Sitter embedding is the only stable embedding. Additional arguments are
given in favour of a previous claim that a class of $f(R)$ models comprising
both positive and negative powers of $R$ (two different mass scales), could be
a nice scenario where to address, in a united picture, both early-time
inflation and late-time accelerated expansion of the universe. The approach
undertaken here is used, also, to check ghost-freedom of a Dirac-Born-Infeld
modification of general relativity previously studied in the bibliography.
