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In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduceda unifying framework to study the learnability of classes of partial concepts.One of the central questions studied in their work is whether the learnabilityof a partial concept class is always inherited from the learnability of some``extension'' of it to a total concept class.  They showed this is not the case for PAC learning but left the problem openfor the stronger notion of online learnability.  We resolve this problem by constructing a class of partial concepts that isonline learnable, but no extension of it to a class of total concepts is onlinelearnable (or even PAC learnable).

Online Learning and Disambiguations of Partial Concept Classes