But more than this, what about all the times when the clearer way to explain an idea, or a simplified proof, or a neat trick to shave off a small constant factor occurs too late to be incorporated into the published version of a proof? You come across a helpful reference too late to mention it (or it is only published after your work is published). The result is that one's "deep understanding" of the problem is never committed to the literature, since none of these little fragments is worth writing up on its own. Even if they get committed to paper, the wealth of knowledge on a subject gets fragmented over a number of papers by a variety of authors.
As a grad student, I struggled against this phenomenon, and tried to produce a thesis that was free of such 'continuity errors'. I carefully worked on the writing to make a write up that collected the results together in a coherent way, and filled in all the gaps and observations that seemed relevant. About a year after finishing it, I realized that I would do it completely differently if I were writing it then. New ideas and new insights arrive constantly, and new ways to communicate ideas and proofs occur with time as one's understanding deepens. Probably one's best hope in this struggle would be to give it enough time and write a textbook or, better still, get someone else to write a textbook. But this is really a futile struggle against entropy.
Returning to the original article, the most practical solution to continuity issues is to enlist a vast army of obsessives to
At this point I received an email from a grad student who had read one of my papers and was asking a very long and detailed but entirely misguided question about a typo in the proof of a fact that was mostly irrelevant to the main part of the argument, and the answer became clear to me.
Errata: thanks also for those who take the trouble to point out my typos and grammatical errors. Distributed checking, maybe that is the answer?