The aim of this paper is to make a contribution to the investigation of the roots and essence of convex analysis, and to the development of the duality formulas of convex calculus. This is done by means of one single method: firstly conify, then work with the calculus of convex cones, which consists of three rules only, and finally deconify. This generates all definitions of convex objects, duality operators, binary operations and duality formulas, all without the usual need to exclude degenerate situations. The duality operator for convex function agrees with the usual one, the Legendre-Fenchel transform, only for proper functions. It has the advantage over the Legendre-Fenchel transform that the duality formula holds for improper convex functions as well. This solves a well-known problem, that has already been considered in Rockafellar's Convex Analysis (R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970). The value of this result is that it leads to the general validity of the formulas of Convex Analysis that depend on the duality formula for convex functions. The approach leads to the systematic inclusion into convex sets of recession directions, and a similar extension for convex functions. The method to construct binary operations given in (ibidem) is formalized, and this leads to some new duality formulas. An existence result for extended solutions of arbitrary convex optimization problems is given. The idea of a similar extension of the duality theory for optimization problems is given.