Noncontinous additive entropies of partitions

We complete a description of additive partition entropies. This time we do not assume I to be continous. In the process we solve a 2-cocycle functional equation for certain subsets of convex cones. 1. Introduction. We assume all definitions and results from [3]. In the current follow-up paper we drop the assumption of the continuity of the additve partition entropy. Our aim is to prove a theorem analogical to Theorem 1 of [3]. As a by-product of our efforts we obtain that all symmetric solutions to a 2-cocycle functional equation for certain subsets of convex cones are coboundaries (Lemma 2, see also Remark 1).