iPhone 4 continues to shut off

I was standing in line yesterday to get food when all of the sudden my iPhone 4 just turned itself off and would not turn back on. Once I got home I plugged it into the charger which was plugged into an outlet and it turned itself back on. It didn't say my battery was low, but all seemed fine. Throughout the day my phone would turn itself off and back on at random intervals even when attached to a charger. Without the charger it would not stay powered on at all. I attempted to reset it to factory settings through iTunes and in the middle of the reset it turned itself off and wouldn't turn back on. Now i've got an empty iPhone that I can't connect to iTunes. This seems like a pretty serious problem with it and I'm afraid that I'm going to have to get a new phone. Can anyone help with this?