Will discontinued apps be avaliable on my iPhone after a software update?

Hi, I'm looking to update my iPhone 4 with iOS 6.0 to iOS 7.1. I have a few apps installed from the apps store that are no longer avaliable to buy, such as the original Flappy Bird. I'm concerned that these apps will not be avaliable after I update. I plan on backing up my phone first, but will the backup save these apps from being gone forever? Is there a way to prevent these apps from being deleted other than backing up my phone? Or will these apps simply be no longer avaliable?

Unless you personally delete them they will stay.

Keep in mind some apps only run on 7 and up