Jaynes mentions a "convenient" continuity assumption following 1.28, and uses it following 1.37. As you point out, the comments following 2.13 seem to indicate something of why Jaynes believes this assumption to be only convenient but not necessary. The comments of talyo just below suggest that Jaynes was wrong - we need something approximating continuity.

But continuity is not difficult to justify. We need only assume that we can flip a coin an arbitrarily large number of times and recall the results. Hmmm. Recall an arbitrarily large (unbounded) quantity of information? Maybe it is not so easy to justify...