Should I Break Up With My Boyfriend?

So I've been dating this guy for 8 months. If I even smile at a another guy (let alone talk to one) he freaks out and completely shuts down. He acts like I don't even exist and he just sits next to me brooding and glaring, he won't talk or touch me either. He's done this several times, for no reason sometimes! We'll be having a good time one moment then I start talking to someone (last time it was his sister) and he gets all mad and again, sits there silently brooding. I told him that if he didn't fix his pissy attitude that I was going home, and then bam! He suddenly cares and wants to hold hands and kiss and make up. I told him to get his hands off of me if he was going to be bipolar but he just kept saying sorry. Like he always does. I told him that this was his last chance several times, but I just can't come around to breaking up with him! Help! What do I do? Should I stay or should I go?