Major US cities restore connectivity locally for a couple of days, but the internet islands are separated (or have very poor interconnectivity). Two days later, connectivity is restored resulting in, say 50 separate blockchains full of transactions. NYC leads the pack with 5% of all transactions and is the dominant chain. How do we reconcile 95% of the transactions (from 49 chains, less than 5% each) ? Would anyone in their right mind use bitcoin in times of trouble?
Most merchants accepting Bitcoin transactions require six confirmations, so if there is no connectivity then, of course, there would not be any confirmations.
So while continuous connectivity isn't required to use bitcoin in commerce, at least sporadic connectivity is.
For merchants who accept on 0/unconfirmed (e.g., retail), the rely on protection from the network to know of a race attack. Without connectivity, that protection is gone and the merchant is vulnerable. If the customer can be trusted, then a transaction on 0/unconfirmed can be accepted even without continuous connectivity.
For when extended outages occur, there wouldn't be multiple blockchains as there's no chance some isolated node (or region of connected nodes) would have the hashing capacity to make it to even one or two confirmations nonetheless six confirmations. So there's no isolated forking issues to be concerned about.
Futher discussion on the suitability of bitcoin on an "disconnected island" occurs here:
An interesting scenario occurs if severance happens when the network very close to a 2016 block.
When mined 2016 block in most powerfully mined network fragment, a new hash rate is computed there -- likely 4x easier than current hash rate. Thus instead of 1 block per 200 minutes (at 5% mining power local to network shard) 1 block per 50 minutes will get mined. The Bitcoin network works locally -- sort of -- adding many blocks before connectivity is restored. Confirmations will require 5x minutes, not 20x minutes.
Now, connectivity is restored. Local chains where the 2016 block was solved are much longer. One of them wins instantly. The whole Bitcoin network accepts lower difficulty and starts operating with a lower hash difficulty until the next 2016 block is solved. It takes blocks 2.5 minutes to solve instead of 10, and it takes nodes 4 times the normal bandwidth to keep pace.
But miners local to the network fragment containing the greatest mining power win. 95% of competition is eliminated, so mining gets 4x easier per block. Blocks take longer, but the earned bitcoin payout per spent mining effort is quadrupled until connectivity is restored.
Difficult imagining though anyone with so much to gain in quadrupled mining payout, that they are willing to cause a global catastrophe.
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.
I can see a close system being very helpful for local businesses. Imagine I want to go to dinner. After eating I pay for it through a terminal connected to the rest of the offline system the restaurant uses as they do now when inputting order, discount's etc. Go to the grocery store and pay for your food at the checkout, (closed system) Yeah, it does nothing for purchasing online, but would it be that difficult to go down to the municipal building to pay your water bill?