How connectedness is related with continuity?

I sent my article Connectors and generalized connectedness to a math journal for peer review and publication.

This my article however does not address an important facet:

It is well known that a set is connected if every function from it to a discrete space is constant.

AFAIK, this holds for topological connectedness and continuity, proximity connectedness and proximal continuity, uniform connectedness and uniform continuity. (Correct me if this is wrong.)

The problem is that I cannot think of generalization for this statement for generalized connectedness with generalized continuity.

Well, see this page to familiarize yourself with both generalized connectedness and generalized continuity as defined by me in the course of my research.

I’m proud that I was able to introduce new concepts of generalized connectedness with generalized continuity into mathematics. But I cannot relate these two. Your comments and ideas are appreciated.