The main purpose of this paper is to investigate constructively the relationship between proximal convergence, uniform sequential convergence and uniform convergence for sequences of mappings between apartness spaces. It is also shown that if the second space satisfies the Efremovic axiom, then proximal convergence preserves strong continuity.

A natural extension theorem for strongly continuous mappings, the morphisms in the category of apartness spaces, is proved constructively.

The paper deals with proximal convergence and Leader's theorem, in the constructive theory of uniform apartness spaces.