Death and the Continuity of Memory

I could write a lot of things about death, but suffice it to say I'm against it. I'm also curious what my own beliefs suggest about what I think constitutes death.

I read a science fiction book a few years ago that I really enjoyed, but unfortunately can't at this moment remember the name or author of. But in this book, you could own a machine that would make clones of yourself, of varying quality (the good ones were very expensive, the menial ones less so). These clones had self-awareness, but they would die (melt down or whatever) at the end of the day. However, you could upload their memories to yourself before they died. (Some people never did, but others, like the book's hero, almost always did.) The book had some kind of a plot involving investigations, and it was told from the perspective of whichever clone (or original) of the protagonist was involved in the action at a given time.

So, you (a clone) would wake up in the morning, and think, "Ah, today I'm the clone." Because you'd have the memories of the person you were cloned from, and that person would have the memories of being a clone as well as a regular person. I think one of the clones once observed, "Some days you're the clone, some days you're the original," which is surely what such a situation would feel like. Some clones were fearful of their deaths, but especially fearful of dying without uploading their memories. To live for nothing!

Sally and I used to talk a lot in college about things like whether using a teleporter (something that destroys your body and builds an identical one in a new location) was the same as death. If I die but an identical person wakes up with the exact same thoughts and memories, what happened?

What about my surgery? When Tam was laid on the table and totally lost consciousness, did she die? And did the body I call my own then wake up later, with me in it?

What about sleep? Will I die tonight, and will someone 'else' wake up with my exact memories?

All of this seems to assume some kind of mind-body dualism, which I don't believe in, but which I also find impossible to get around in my own head.

But what seems clear is that continuity of memory is the key to not feeling as though you've died (or that someone else has died and been replaced by you, I guess I should say). When that clone wakes up in the morning, he remembers being a clone before, many times, and that makes him feel like he won't die either. I'm not afraid of dying from falling asleep, because I remember falling asleep many times and I'm still here. And if I went through a teleporter a few times, I'd lose my fear of dying from it, because I would remember having done it and come out the other side.

Is that enough?