To continue an activity is to keep it going, either with or without interruption. If you ask the newspaper to continue delivering your paper while you are on vacation, you want to receive all the news while you’re away.
The verb continue is related to the word continuous, from the Latin word continuare, meaning “join together” or “connect.” When anything goes on without a break, like the middle school variety show, it continues, uninterrupted. You can also continue something that was paused or set aside. You could, for instance, continue watching the movie that you paused when the pizza was delivered and then continue to drive your parents crazy by texting your friends while watching the movie with them.