Short answer: Yes.
Do I have to loose my email address if I change providers?
No, not usually. Most providers have a service that lets you keep your email address with them while getting your core internet service from someone else.
This was a problem that people moving from dial up to ADSL faced in the 2000′s. It’s not a new problem and just one you need to talk to your existing and new providers about.
It’s also a problem that people moving from one ADSL provider to another face all the time.
Many people choose to use services like Hotmail and Gmail just so they can take advantage of ISP deals without having the hassle of being tied to an email address.
You can also set up your own domain name and have your own email addresses for your whole family without being attached to a telephone company.