alt Hacker NewsCell phone towers and communication systems have backup power for emergency communication during power outages.
If you have backup power for your router and ONT/Modem, you should also still have internet service during a power outage. The ISP-owned ONT for a place I lived had a little lead-acid battery attached to it, and during power outages I still had internet service.
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I live somewhere with two nines of utility power reliability, I mention that to say everybody's backup (or lack thereof) is well tested. I've got UPSes on most of my computers and a standby generator that pops on after about 10 seconds. DSL from my ILEC has zero backup power; sync drops when the power drops. I don't know about the cable. Municipal fiber doesn't drop so far, but I haven't had a long outage since I got it; my ISP has a generator where they route customer packets. We get cell coverage for about 4-8 hours, depending on which network you're on and if the outage started overnight coverage usually lasts until people wake up.
After that, communicating with the outside world is hard for most people. Time to make babies ... anyway it's often cold, so snuggling is likely.
ACT terminates the fibre outside (possibly terminating a few services at once), and only spec their backup battery for 1–1½ hour (I have observed about 1½ hours on what should be a fresh battery).
Other ISPs I’ve used in Hyderabad (Hireach, Airtel) terminate the fibre inside, so you can put whatever backup you like—I have a small ₹1,000 thing which kept the Hireach one going for around 4½ hours, and the Airtel one for 1¾ hours. (Still haven’t got a proper inverter, not so convenient when renting.) The connection always worked until my battery ran out.