Connecting early and late universe by $f(R)$ gravity
  Inflation and dark energy are two of the most relevant aspects of modern
cosmology. These different epochs provide the universe is passing through
accelerated phases soon after the Big-Bang and at present stage of its
evolution. In this review paper, we discuss that both eras can be, in
principle, described by a geometric picture, under the standard of $f(R)$
gravity. We give the fundamental physics motivations and outline the main
ingredients of $f(R)$ inflation, quintessence and cosmography. This wants to be
a quick summary of $f(R)$ paradigm without claiming of completeness.
