How does the user log in if they've changed their email address (e.g. switched ISPs and not using a gmail address)? Authenticate using SMS?
That is the reason, you should try not to change your eMail to often and when, keep the old adress until it gets cold.
Of course, what you normally should do: Change your eMail-address in the application before you switch.
AND: Never use your ISPs eMail-address (unless he will let you use it after you canceled your contract).
The system will probably have to give a way for the user to change email address while the user still has access to the old email address.
This is something that wasn't mentioned in the article but should definitely be solved.
Any reasonable web-application today has (or should have) a possibility to change eMail address.