Topologically inseparable functions II: Infinitary case

Given a set A and a function A: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f ⁿ , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A ⁿ → A.