I am naively asking this with real interest because I have no idea: Can it continue to function at any price? Is there a price so low that new transactions would not processed?
The only extreme possibility would be a temporary denial of service if hashrate dropped too quickly, causing blocks to come extremely slowly until the network adjusted to match the current hashrate, resetting block time back to the target.
In such an extreme scenario, this could effectively pause commerce for months if nobody helped bring the hashrate back up until the adjustment.
There are many mitigations possible:
1. The remaining miners have newfound control of the network so could push in a difficulty adjustment algorithm change to something more modern and adaptive, restoring service immediately. This has been done on so many other smaller networks that it isnt experimental.
2. The remaining miners could do other things too like a one time sooner difficulty adjustment
There are a few networks experiencing this right now and addressing it in different ways. There are always a few real money cryptocurrency networks going through these growing pains so we can see all theoretical experiments happen extremely fast.
Now please correct me if I'm wrong, but IIRC the difficulty adjustment requires some blocks to get mined, so if the hash rate would drop extremely quickly, the duration needed to adjust to the lower hash rate could take a long time. However that would mean the potential block fees would be steadily increasing so maybe it would quickly fix itself (make mining those blocks more profitable).
That said, the "price" (in USD) is a function of supply and demand and is completely disconnected from this. The price of a BTC could drop to $1, and mining would still work fine.
only after the difficulty adjusts downward
which could be as long as 2016 blocks