Edge Elements (EE) have received in recent times much attention from the Finite Element community. The present contribution analyzes the role played by EE in the preservation of the fundamental prop-erties of physical equations submitted to discretization. A short review of these properties is pre-sented, where it is emphasized the presence in the equations, of intrinsically discrete terms that can be represented in a most natural way using the concept of cochain. It is then shown how EE are instru-mental in the bridging of the gap represented by terms which cannot be exactly discretized. It is main-tained that the role of EE lies in their ability to provide a simple machinery to build a continuous rep-resentation (ideally, as a differential form) for a field starting from its discrete representation in terms of a cochain, an operation for which the name of cochain-based field function approximation is sug-gested. The interpolation practices of Finite Elements and Finite Volumes are considered under this light to clear away some confusion and show the way to further generalizations