I have DSL.
My power went out earlier this week for about 6 hours.
But my phone still worked, so I rigged up stuff to get power to my DSL box so I could get online.
Nix, nada, zip. Even though the phone was working, the DSL subcomponent would not connect.
So there ust be some relay type equipment upstream used by the DSL stuff that runs off of the public power, so it was out also.
DSL still uses the same copper POTS did, and if you didn't switch to VoIP, that's why your phones (the hard wired ones) still work. If you switch to VoIP, then your phones will be down when the power goes out because they need power just like you DSL modem needs power.