My iPhone will not sync email with time Warner cable. They say- mail.twc.com

That is a question that you need to ask time warner.

Apple does not handle the servers used for roadrunner mail. You will need to contact your email service provider to collect the proper email settings. You can use the following link to access the article that will help you along

http://support.apple.com/kb/TS3899?viewlocale=en_US

if the link does not show then use the search bar at apple.com/support to search for TS3899

You're not using apple mail server so it's not their issue. You're using time warner or roadrunner it's their issue.

The instructions are at: https://www.timewarnercable.com/en/support/internet/step-by-step/twc-mail.html

If you can't send emails when you're not using Road Runner internet try using the SMTP server: mobile-smtp.roadrunner.com

The instructions are on Time Warner's website but it will work for all roadrunner.com email addresses. Road Runner was a subsidiary of Time Warner and in 2012 they renamed Road Runner to Time Warner Cable Internet. You can read about their history and other companies that offer Road Runner internet at:

https://en.wikipedia.org/wiki/Road_Runner_Internet

iPhone/iPad