Life is really not like the movies. When relationships end most of the time they end. The boy and girl do not decide to be best friends forever. There are some instances when that happens, but it is not very often. If it does happen it is often after a period of healing and reflection has allowed the sting of the break up to pass.
Often times when people split up they try to remain friends. This is often not possible because jealousy or animosity can get in the way. But why in the world or how your friendship can last when everything you had in your relationship just broke down. So when an ex ask that question it might only be polite to say “sure why not.” But deep inside both of you know that it might be hard to be good friends again. That expression is just used as a consolation prize for the dumping you had. This can be even worse when both parties are too proud to admit that they really shouldn’t be friends.
How can both of you be friends if you are not able to go out, talk or have fun most of the time. A friend is a person you develop your trust through time. What happens during that time is that you can become comfortable again, and then your feelings return. When this happens you may slip back into the relationship, and this is really not something that you should allow to happen. You broke up for a reason. It is time to face that, and stay away from each other.