  A generic feature of viable $F(R)$ gravity is investigated: It is
demonstrated that during the matter dominated era the large frequency
oscillations of the effective dark energy may influence the behavior of higher
derivatives of the Hubble parameter with the risk to produce some singular
unphysical solutions at high redshift. This behavior is explicitly analyzed for
realistic $F(R)$ models, in particular, exponential gravity and a power form
model. To stabilize such oscillations, we consider the additional modification
of the models via a correction term which does not destroy the viability
properties. A detailed analysis on the future evolution of the universe and the
evolution history of the growth index of the matter density perturbations are
performed. Furthermore, we explore two applications of exponential gravity to
the inflationary scenario. We show how it is possible to obtain different
numbers of $e$-folds during the early-time acceleration by making different
choices of the model parameters in the presence of ultrarelativistic matter,
which destabilizes inflation and eventually leads to the exit from the
inflationary stage. We execute the numerical analysis of inflation in two
viable exponential gravity models. It is proved that at the end of the
inflation, the effective energy density and curvature of the universe decrease
and thus a unified description between inflation and the $\Lambda$CDM-like dark
energy dominated era can be realized.
