Common sense should prevail. When it comes down to it people will come and go in your lifetime, but good friends are the ones that last. Having a good friendship requires a certain mix and sometimes one's mental health can affect that.

Obviously in a perfect world we could just just answer this question with an obvious no of course no one should give up on you because you have a problem you haven't finished tackling yet, but that's just it, sometimes people really aren't doing anything about it.

I have high standards for people and it does take a certain something to stay as my friend. While having a mental illness does not mean I'll give up on a person, I will if it seems to me like they are a lost cause, refuse to truly deal with their issues, and/or simply don't fit my standards for a friend.