On the continuity of residuals of triangular norms

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