  Perturbative techniques are important for modified theories of gravity since
they allow to calculate deviations from General Relativity without recurring to
exact solutions, which can be difficult to find. When applied to models such as
$f(R)$ gravity, these techniques introduce corrections in the field equations
that involve higher order derivatives. Such corrections must be handled
carefully to have a well defined perturbative scheme, and this can be achieved
through the method of perturbative constraints, where the coefficient of the
additional term in the action is used as expansion parameter for the quantities
of interest. In this work, we implement a perturbative framework that compares
solutions in modified theories of gravity with solutions of the Einstein field
equations, by following the guidelines of perturbation theory constructed in
General Relativity together with the perturbative constraints rationale. By
using this formalism, we demonstrate that a consistent $f(R)$ perturbation
theory in vacuum, for an important class of $f(R)$ functions, produces no
additional effects with respect to what is expected from the perturbation
theory of General Relativity. From this result, we argue that there are
fundamental limitations that explain why the solutions of some $f(R)$ models
can be disconnected from their general relativistic counterparts, in the sense
that the limit that leads from the $f(R)$ action to General Relativity does not
transform the solutions accordingly.
