stamina mode

[Matu]

How to make it so it doesnt cut off the internet connection

[_alexdon_]

You cannot,that is its purpose.

[Hampus]

Just like alexdon said, this is the purpose of STAMINA mode. You can read more about STAMINA mode at the following link:
http://talk.sonymobile.com/t5/FAQ/Battery-time-and-enhancement-features/m-p/347874

[_alexdon_]

You could use low battery mode instead,you can customize it.