Discontinuity

I might stop mentally reprocessing the basic paradox of the Time Patrol series. On the other hand, it is a basic paradox which means that some minds continually reprocess it.

The only meaning that I can think of for "discontinuity" is as follows -

There is a single timeline in which some time travelers (i) arrive from the actual future with memories of that future whereas others (ii) merely appear with memories of prevented futures but (I think that) there is no third category of time travelers who (iii) both arrive from an actual future and find on arrival that that future has been prevented.

(i) is logically possible;
(ii) is counter-intuitive but nevertheless logically possible;
(iii) seems to me to be logically impossible.

If one future is both actual and prevented, then two timelines are necessary and they are in a before-and-after relationship along a second temporal dimension. This is staircase thinking. I walk or swim and the problem reformulates itself but it can't (?) go on forever.