Some remarks on approximations of the Zadeh's extension

Within this contribution we study the problem of approximating the Zadeh's extension of a given function. Among other things, we showed that the technique of F-transform can effectively approximate this extension. We also studied a general problem of the choice of the metric on the space of fuzzy sets (resp. numbers) and we showed that, in comparison to recent habits, the choice of the levelwise metric need not be the best one, and that the approximation process can be simplified. We also slightly discussed the ability of fuzzy rule based systems to approximate the Zadeh's extension. Our results are general in the sense we mostly require nothing but the continuity of the original map.