Should we just break up?

My boyfriend keeps questioning why I even like him, saying that no woman would ever like a guy that is in his shoes since he is in a weird place in life. And he keeps telling me that I will never be happy with him, that I could have any guy, so why choose him.
After a big fight a few weeks ago, I realized that he was being a jerk and was picking fights all the time, so I told him we needed to break up. He begged me to let him try and fix our relationship, to give him a second chance. So I did, and he is being nicer, but now he is pushing me away by constantly questioning why I like him. He even is questioning my judgement on my education which is getting very annoying. He is now saying that it doesn't even feel like we're really dating.
I honestly just don't think I can deal with this for much longer... Should I just end it?
Should we just break up?
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