On the continuity of residuals of triangular norms

In this work a long-standing problem related to the continuity of RR-implications, i.e., implications obtained as the residuum of tt-norms, has been solved. A complete characterization of the class of continuous RR-implications obtained from any arbitrary tt-norm is given. In particular, it is shown that an RR-implication ITIT is continuous if and only if TT is a nilpotent tt-norm. Using this result, the exact intersection between the continuous subsets of RR-implications and (S,N)(S,N)-implications has been determined, by showing that the only continuous (S,N)(S,N)-implication that is also an RR-implication obtained from any tt-norm, not necessarily left-continuous, is the Łukasiewicz implication up to an isomorphism.