Then, we learn as the play progresses that two of the characters, The Boy and The Child who have no lines, are dead all along and we actually get to see them relive their deaths near the end of the play. So, if these children were dead all along but then act out their deaths, what does that make them to begin with: dead or alive? It is my own understanding that perhaps none of these characters are really alive. I imagine them to have literally walked out from the pages in which they are written - sort of in an Edenics way where the words produce the actual thing. If that is the case with this family, then it's reasonable to argue that none of them are really real - they are just manifestations of the words and they popped into being. They have no history in our world as real beings, The Mother did not physically give birth to her children nor her mother before her and so on. If none of these characters are really alive, then none of them can technically die as The Father has previously stated. So, by that logic the children don't really die in the end of the play when they reenact their scene because they were never really alive or real to begin with.
Also, the part of the play that was written, or even just their creation as an idea, makes them immortal. Think of every book you've ever read. Do those characters die in your imagination just because you closed the cover at the end of the story? No! You continue to imagine them and reference them for as long as you can remember the story; you go to class, a book group, or chat with a friend about the story which makes the characters alive beyond the boundary of the pages. And, for generations people will read the same books, passing on the characters and their adventures for decades or more.
This, obviously, isn't the only or necessarily correct analysis of this play so I would love to hear what other people think.