I don't see the point in trying to be friends with someone after they hurt you (or you hurt them) emotionally, which is usually what happens...sour feelings with the other person after you break up. I don't think there's many relationships out there that end where both people are perfectly happy moving on from each other and rainbows, cupids, and butterflies shoot out everywhere and there's this perfectly happy ending with the two people going their separate ways.
There would have to be a reason to want to continue having them in your life after everything is said and done. One is you (or the other person) wants to continue being in each others lives for whatever reason...to always keep you on the back burner as a back up, or who knows. Another thing might be "ohhh yeah we are friends now, so I don't necessarily need to commit to you per say, but I still wouldn't mind getting with you every now and then (friends with benefits)." I've never been down for that and I never will be.
There are certain kinds of people who may think that way, but most of the people I know wouldn't do that.

One of my best friends is an ex. We dated in 2008 for 3 months and I ended the relationship because he wasn't right for me. He's an incredible, kind, sensitive person and he asked if we could stay friends. We met a week or so after we had broken up, which was a little strange, but after that, everything was fine. He and I both feel that our relationship led to a really good friendship. He and I go to one another for advice in the same way two long-term friends would - we have been friends for 5 years, so that's only natural!

I used to talk to all but one of my exes; I didn't talk to one because we had dated for a month and we had nothing in common. I recently cut ties to four of my exes for different reasons. However, it had nothing to do with the fact we used to date.
For example, I cut ties with my first boyfriend because he was a terrible friend.

I am at a point in my life where I want to focus on positive relationships. If friends go through anything rough, I will be there for them, but this 'friend' had been mildly abusive for years. I don't have time for something like that, ex or not.