I wouldn't be too sanguine about being able to use the internet during a power cut - unless it's just your property, the likelihood will be that some upstream network device in your area will also be taken out once the power goes, rendering your power supply moot.
In the past when we have had power cuts in our street (there was a dodgy transformer locally, since replaced), the Virgin media boxes used also go down. Virgin used to send a man with a van to power up the voice side (as they were legally obliged to do so), but not the data side, so even though I could power the cable modem with a UPS, the upstream connection was dead. Then when the power came back, more often than not, a network card in the street cabinet would have tripped / failed, requiring Virgin to send a repair team out to replace it.