What happens to my monitored alarm if the power or internet goes down?

The alarm panel runs on a standby battery, so it keeps working through a power cut for a period set by its grade. Signalling depends on the path: a mobile path usually survives a broadband failure and vice versa, and on a supervised path the Alarm Receiving Centre is told when a path is lost rather than simply hearing nothing.

Standby battery duration is a graded requirement, not a manufacturer whim, which is one of the practical consequences of system grade. It is finite, so a multi-day outage is a real limit worth knowing about.

A digital voice line dies with your router, which is exactly why landline-based signalling is no longer appropriate. A mobile path has its own power inside the communicator and its own network, so it is independent of your broadband and your electricity in a way a router-based path is not.

The supervision point is the one people miss. On a properly configured monitored system, silence is itself information: if the ARC stops being able to poll the system, that is an event. Ask what the detection time is for your configuration and what the ARC does when it triggers.