All telephone modems have (by law) a battery backup for the telephone, but NOT for the internet. "Ah ha", you think, "I'll just put the modem on a UPS!" But it turns out that the "telephony modem" will not charge the battery fully if the power source is on (some) UPS.
Has anyone found a way to keep the internet connection going during a power outage and STILL charge the telephone battery backup?