Strong uniform continuity

Abstract

Let $\mathcal{B}$ be an ideal of subsets of a metric space $〈X,d〉$. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of $\mathcal{B}$ which reduces to ordinary continuity when $\mathcal{B}$ consists of the finite subsets of X and agrees with uniform continuity on members of $\mathcal{B}$ when $\mathcal{B}$ is either the power set of X or the family of compact subsets of X. The paper also presents 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 a net of continuous functions.