Any decent (by definition) DMM will survive mains voltage on it's input terminals when in "continuity" mode, or in almost any other setting of the dial.
The only exception is the Amps input terminals. It is a bad Idea to try to measure how much amps a mains socket can deliver.
Usually the fuse blows if you try this.
The best way to pevent this is to buy a DMM that beeps at you when the banana plug is in the amp socket, while the dial is not in a current measurement mode.
This has saved me a few times (and the DMM's I was using.
My weirdest failure yet was ... 30+ years ago, when I was still a young brat.
The DMM I had then was an unfused el cheapo and I did try to measure amps out of a mains socket.
Lot's of sparks and noise, scared the hell out of me.
After that the DMM did not work properly anymore, even thou it sometimes gave somewhat plausible readings.
Or, at least I thought so.
I put the meter aside as I found it logical it would not survive such an abuse.
Some time later, I examine the meter a bit closer and discovered that the test leads had melted. After replacing them the meter still worked OK-ish in most settings.