You may find that when you enter your existing email address into your new ISP it will reject it as it is already in use. (by you of course). If that happens then unfortunately you can't keep it. You will be offered some alternatives, generally with some figures on the end. Bit of a bind but if you can't keep your existing address you would have to send an email to everyone in your address list advising them of the new email address.
You are partly correct. In my experience, if you have a Pay As You Go account with your ISP, you should be able to keep your email address even though they are not your "main" ISP any more. Some require you to dial in every so often ,some don't.
If however, you have an unmetered account with an ISP which you terminate, they will probably cancel your email account as well.
Examples, Tiscali, ic24 Pay Go accounts no requirement to dial in.
Virgin, say they require you to dial in every 60 days but do not enforce this.
(The poster doesn't say which type of account they have currently).
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