Meanwhile, there are a few different ways to keep your resolv.conf file untouched, rather than relying on isc-dhcp-client to continually rewrite it in the form you want. The wiki page describes some of those ways. Personally, I do not understand the appeal of the "put lines in configuration file X so that isc-dhcp-client will use them when it rewrites configuration file Y" approach. I'd rather just edit file Y by hand and tell isc-dhcp-client not to touch it at all. It's a shame that it's so incredibly difficult to do that. But, that's why we have the resolvconf package, and it's why we have the wiki page that describes how to do it.
Just edit the file /etc/resolv.conf and make it immutable (chattr
+i /etc/resolv.conf). At least you will know what is in the file
and that it can't be changed (mistakes and all). I use this to
keep chrome from changing the google-chrome-beta.list file every
time it starts since I use the HTTPS/// option of apt-cacher-ng
to use https transfers outside my LAN.