  We address the question of how to build a class of $f(R)$ extensions of
General Relativity which are compatible with solar system experiments, without
making any preliminary assumption on the properties of $f$. The aim is reached
by perturbatively solving the modified Einstein equations around a
Schwarzschild background and retrieving a posteriori the corresponding $f(R)$.
This turns out to be non analytical in $R=0$ and should be intended as the
leading correction to the Einstein-Hilbert action in the low curvature limit.
The parameters characterizing the $f(R)$ class are then set by constraining the
corrections to four different local tests with the observations. The result is
a class of $f(R)$ theories built up from a purely bottom-up approach and
compatible with the local tests. At a more general level, this result can help
constraining exact $f(R)$ models working in Cosmology, since it provides the
correct local limit. Further developments and possible extensions of the
approach to Cosmology are also discussed.
