It depends on the situation. If someone did something terrible, then I doubt that a friendship would work out very well between the couple after they break up. But if it's a mutual breakup then I see absolutely no harm in staying friends. There's no need to throw something good away just because it didn't work out romantically between them. Personally I'm still friends with all of my ex's.
Its different for everyone. I have been friends with an ex boyfriend for over 30 years. I am friends with my ex husband. He has dinners here with the whole family.
Still, if someone is not comfortable with that relationship for themselves, then no. Sometimes the emotions are just too strong.
I believe that if 2 people have kids out of the relationship that they should think about the kids and if they love the kids very much then they should be friends for the kid's sake. So kids can grow up watching the behavior of both parents and learn that we shouldn't be enemies there is too many of that out in the world. If both people have no kids then they don't have to be friends if they don't want to but they also don't have to be enemies too.
I don't think I consider my ex husband my BEST friend, but certainly we are friendly to each other. Sometimes the way he treated me still gets my blood boiling, but he is not a bad person in himself. Because he is such a good father, I look on him with kindness.
I know a freind of mine was still very good friends with her ex-husband. She never really wanted the divorce, but consented to it. They had a very good relationship afterwards for their 2 kids. She remarried and he also remarried and has more children. She still got along with him and his new wife. He recently died and she was so heartbroken, her current hubby didn't understand.
Let some time go by, and you can often be friends again IF the relationship was basically a good one and the reasons you brok up were just circumstances or wanting different things in life, not cruelty, lies, abuse, or anything of that nature.
Also if you're married, I think you have to include the spouses in the friendship, not make it a one-on-one sort of thing.
I'm still very good friends with my most serious ex-boyfriend (and his wife), from h.s. and college days; that relationship goes back more than 40 years. It's nice for both of us to have that "someone who knew you when" in our lives at this age.
I dont know if my friendship with one of my friends will survive after she had an affair with one of my good friend's husband and now is pregnant with his child. I know that my good friend hates her with a passion and their friendship is ruined but my friendship with her is also in question because what she is doing is making me sick to my stomach now.