My best friend gives me mixed signals constantly, should I give up on him?

After telling my best friend I liked him he was off and on. He can easily change his mind. A few nights ago we were with family friends, being flirty even after we decided we were going to be just friend, he decides to give me his jacket because I was cold. We kept making eye contact, flirty, and him touching my arm constantly. After we went to go watch a movie and he sat beside me. He was trying to hold my hand but I didn't want to at first because I thought we were just friends(even if I didn't want to be), so he then puts his hand on my leg. It didn't last long, but he kept doing it. Then he tired to hold my hand again and I let him. A few minutes later he let go and moved away from me. He tried to avoid eye contact. When we left there was no good-byes or anything of the sort. I'm unsure if I should still be interested in him or if I should just give up liking him or even give up being friends with him.