Abstract:  Let B be an ideal of subsets of a metric space < X, d >. We consider a strengthening of the notion of uniform continuity of a function restricted to members of B which reduces to ordinary continuity when B consists of the finite subsets of X and agrees with uniform continuity when B is either the power set of X or the family of compact subsets of X. We shall present also new function space topologies that are well-suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of the net of continuous functions.