How can a person find corrupted software?

I have an Iphone 5 and for some time now, I have not had any kind of cellular connection. When connected to WiFi, the phone works just fine. Any other time, Safari or any number of apps that require an internet connection will not work. I took the phone to an Apple store. They said that all of the hardware was fine and this it was a software issue. Erasing all the data to factory defaults enabled the cellular connection to work. Once I restored my data from iTunes, the problem returns. I would prefer not to start from scratch and permanently delete all my data. If any one knows how to determine if an app is not a good one or how to fix Safari/cellular connection, it would be greatly appreciated!