The Δf(r) descriptor characterizes the variations of the absolute hardness when the external potential changes. In this work, the Δf(r) descriptor is regarded through three different theoretical points of view: its relations with the Fukui functions; its links with the covalent interaction energy; and finally its compliance with the Principle of Maximum Hardness. It appears from the three studies that the sign of the descriptor is able to characterize the electrophilic/nucleophilic behavior of a molecular site.