Il concetto di funzione. aspetti epistemologici e didattici

We recall the main steps of the historical process which brought to the modern concept of function. We face some didactical questions related to its learning, such as: the question of the control of its different conceptions appeared over the years; the limits of the persistence of an old language; the conflict between definitions and images of a concept; the question of the coordination among different registers of representations. We then indicate the new trends in its teaching and present some results concerning an innovative path, experimented at compulsory school level, aimed at exploring binary relationships between variables and at studying simultaneously the pair <function, inverse function> in the case of 1-1 correspondence.