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A generic feature of viable $F(R)$ gravity is investigated: It isdemonstrated that during the matter dominated era the large frequencyoscillations of the effective dark energy may influence the behavior of higherderivatives of the Hubble parameter with the risk to produce some singularunphysical solutions at high redshift. This behavior is explicitly analyzed forrealistic $F(R)$ models, in particular, exponential gravity and a power formmodel. To stabilize such oscillations, we consider the additional modificationof the models via a correction term which does not destroy the viabilityproperties. A detailed analysis on the future evolution of the universe and theevolution history of the growth index of the matter density perturbations areperformed. Furthermore, we explore two applications of exponential gravity tothe inflationary scenario. We show how it is possible to obtain differentnumbers of $e$-folds during the early-time acceleration by making differentchoices of the model parameters in the presence of ultrarelativistic matter,which destabilizes inflation and eventually leads to the exit from theinflationary stage. We execute the numerical analysis of inflation in twoviable exponential gravity models. It is proved that at the end of theinflation, the effective energy density and curvature of the universe decreaseand thus a unified description between inflation and the $\Lambda$CDM-like darkenergy dominated era can be realized.

Cosmic history of viable exponential gravity: Equation of state oscillations and growth index from inflation to dark energy era