If an ISP link physically shows down on your device, and the device supports management, often this can be monitored. Various ways to do that, much depends on what the network device supports, and how/what will be doing the monitoring.
That said, with ISP links, a physical link might go down, not attached to any of your devices, or the ISP has some other issue that precludes using their service (i.e. you cannot reach anything when you attempt to).
For those, you need something that actively "probes" your ISP is working, and if not, you can detect that. Again, various ways to accomplish that. Also again, much depends on your equipment.
Some of those products you can run on a PC, and they will actively, and repeatedly, try to contact other destinations, and can "alarm" in various ways, if one or more of their "targets" cannot be reached.
Lastly, sometimes you can also use a hybrid approach, taking advantage of what a network device, itself, can tell you, and what you can determine with a program attempting to "use" the network.
(edit - PS:)
BTW, laugh, most networks already have "built-in" ISP monitoring. I.e. your phone rings and caller says, I cannot reach www.xxx.com. You hang-up, immediately phone rings again. That caller says, I cannot reach www.yyy.com. You hang-up, immediately phone rings . . .
