  We study the relation between Minimal Dilatonic Gravity (MDG) and f(R)
theories of gravity and establish strict conditions for their {\em global}
equivalence. Such equivalence takes place only for a certain class of
cosmological potentials, dubbed here {\em withholding potentials}, since they
prevent change of the sign of dilaton $\Phi$. The withholding property ensures
the attractive character of gravity, as well as absence of ghosts and a tachyon
in the gravi-dilaton sector and yields certain asymptotic of the admissible
functions $f(R)$. Large classes of withholding cosmological potentials and
functions $f(R)$ are found and described in detail. It is shown that the
popular choices of $f(R)$ functions are not withholding ones. The particle
content of the gravi-dilaton sector is found using perturbation theory around
de Sitter vacuum of MDG. The graviton remains massless, since it obeys
conformal invariant field equation in the de Sitter space-time. The $R/6$ term
in the conformal invariant wave operator introduces a very small mass scale
$m_{R} \approx 1.5 \times 10^{-38} m_e$, $m_e$ being the mass of the electron.
The mass of the dilaton is much larger: $m\gtrapprox 10^{29} m_{R}$. Two new
phenomena: scalaron waves and induction of gravitational waves by the scalaron
field are discussed using the derived wave equations for scalaron and graviton.
The MDG and f(R) theories are shown to predict physical deviations from GR.
Seemingly, the MDG and f(R) theories, when globally equivalent, offer a unified
description of dark energy and dark matter.
