carry on

(idiomatic) To have an illicit sexual relationship.

Example:   I'll be gone for a few days, but I hope you will carry on in my absence.
To take baggage or luggage onto an airplane, rather than check it.
  You may only carry on items that are smaller than a certain size.
  It is difficult to carry on a conversation with so many distractions.
  to carry on commerce in a market
  I really wish you wouldn't carry on like that in public!
  I thought he was my friend, but all the time he was carrying on with my wife!