Around here it doesn't matter if they have modems with batteries or a large UPS for it, when the power goes out the cable and internet goes with it, i know because my modem and computer are on a large UPS, and when the power drops the entire cable system goes with it.
I have been looking at the Verizon home phone connect, its a small cellular phone and battery in a little router sized box with an antenna and a phone jack to plug in regular phones or the entire house in place of the landline at your demarc. I know a few people that have these now, they unplugged the house line in the demarc box and then went inside and plugged the Verizon home phone connect to a phone jack where it has a good signal and now all the wired phones in the house have service through it. Works great.
I will probably get one for my kids to have phone service for emergencies, i don't want to deal with them losing a cell phone, damaging one, running up the bill, etc. I only have cell phones here, no landline phone. When the power goes down the cell tower here stays up thanks to a large UPS and generator on site, when we got wiped out by a tornado in 2011 the only thing that worked was the cellular service.