Unfortunately, your "the fact is" is incorrect. They don't have visual sight of the sun. But they still have day. Because, as I've explained and is an accepted fact, the day is a period of time based on the sun. In the past, before we had recorded a calender based on the sun's movements, they would have been screwed and not known what time it was. They did not know whether it was day nor night.
As for you saying that the periods of time only work if there is a sun or moon to show a clear line between them... Yes. That's... Kind of what I'm saying. The difference is that you seem to insist that this can only apply with visual sight of one or the other, which is simply not true.
Now, we know it regardless of where we are in the world. We always know where one or the other is. We've set up a time region for every portion of the earth based on the sun, and everyone knows whether it's day or night, regardless of whether they can see the sun or not, because after examining astrology for so long, we know where the sun is regardless of visual sight. On top of that, there will always be pats of the earth that see the sun, so they can share such information anyways, and with timezones, that itself can be used to tell the difference between day and night at any point of the earth.