  In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced
a unifying framework to study the learnability of classes of partial concepts.
One of the central questions studied in their work is whether the learnability
of 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 open
for the stronger notion of online learnability.
  We resolve this problem by constructing a class of partial concepts that is
online learnable, but no extension of it to a class of total concepts is online
learnable (or even PAC learnable).
