Debian based OS's have a similar issue as the behavior in RHEL/CentOS dhclient, which is that if you have an interface that relies on DHCP, if the first attempt fails, it will quit and stop. This is a problem especially if you are using your Linux as a router or something else mission critical, but where the internet for some reason may have been down or the DHCP server it gets a lease from broken.
The expected behavior you would hope is that when things are back online that the device will get a lease, but this is not the default behavior. The default is to quit dhclient.
The Debian/Mint/Ubuntu Solution
Fortunately you can just edit /etc/network/interfaces and add this line for your NIC (assuming it is eth0):
allow-hotplug will give us the desired behavior, you can test it yourself that even if the NIC is offline or internet is down, dhclient will still be running for the interfaces specified under allow-hotplug.
This is the equivalent solution to the RHEL/Centos DHCP Persistent Solution
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