50 different views of the current state of the network develop. I'm going to assume that 5% of mining power is also in NYC. What happens is that when the network problems are lifted, the NYC blockchain instantly "wins". Every other miner loses all of their work. Consequently, there is zero reason to mine unless you will win the race.
However, a funny thing happens. People in every non-dominant region have an incentive to connect to a well-informed region to find out about new blocks. A satellite connection of about 10 Kbps is enough to keep up with the network. Once they have mined their block, they might as well tell local merchants about it, keeping them in sync too.
But let's suppose for a moment that satellite internet is entirely out of the question. Would a station wagon full of tapes driving down the highway be enough to keep your miner in sync? Not really, unless you're very close to a well-informed node. However, it might be worth it to somebody who wanted to get their transactions in or check on their transactions.
Would anyone in their right mind use bitcoin in times of trouble?
That depends on the disaster. If this is a freak series of coincidences that caused all of these redundant systems to stop working all at once, then I think everybody is going to wait. What's harder, trying to figure out whether you could get scammed, or closing shop for 24 hours?
On the other hand, if this were an alien invasion that was here to stay, I think people would continue to use bitcoin; however, they'd fall back more often on trust based systems. They might refuse to deal with new customers unless they were recommended by an old customer. Occasional station wagons would keep merchants reasonably in sync.
2That's a very good question. – gosmond – 2013-03-21T05:07:52.207
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possible duplicate of What would happen if a portion of the Bitcoin network was separated from the rest of the network?
– Murch – 2013-09-04T23:04:39.5631
possible duplicate of Countrywide Internet isolation, inevitable fork
– cdecker – 2013-09-05T11:17:23.827