It seems to me that the only definition of "self" worth its salt is one in which the individual can tell who she or he is, not one in which others can tell who you are (which is impossible).
How is that impossible? Everyone does that all the time.
By your definition, both people are the same person at one point
Small but important correction: Two instances of the same person.
but then mysteriously they "diverge" into two people.
Where's the mystery? They have different experiences, so they change.
At what point do they become two people instead of one?
They are two instances of the same person. At an entirely arbitrary point, we may decide to regard them as two different people.
Do these two people who are the same person EVER feel that they themselves are the same person?
What does that mean?
If you mean, do they share each other's subjective or objective experiences, then no, that's absurd.
We all define ourselves by using continuity of time and space.
Not really. We define ourselves by continuity of memory more than anything else. However, in the real world that goes hand in hand with continuity in space-time. (Our experience of time is of course discontinuous.)
If we were to all of a sudden unexpectedly jump into another person's body, or if we were to look in the mirror and have a different face than we expected to see there, we would have a serious identity crisis. After some amount of time, we would get used to the idea and probably begin to identify with the new person.
Erm. If you say so.
Using the same notion of continuity, even the copy knows that she/he isn't the original.
If there is evidence of the discontinuity, yes. If not, then the copy
has no way of telling.
The original knows before teleporting that if he is suddenly in a different location (i.e. if the teleportation seemed to "work") then he is really a copy of the original.
You can find yourself in a different location without being copied. That alone is insufficient, as I have pointed out repeatedly.
If he remains in the original location (i.e. if it failed) then he is the original.
Or not. It could be the copy, while the original has been whisked off somewhere else.
The only way to tell is perpetual surveillance.
If he finds himself dead, then he would realize what a moron he is for killing himself thinking that he would somehow be the copy simply because he defines the copy to be "the same" as him!
Well, no. He won't realise anything at all. He's dead.
It sounds like a lose-lose proposition to me. Certainly in no case does the teleportation really have the desired outcome (actually teleporting from point "A" to point "B" without death).
Only because that is your definition of "death". For other definitions of death, this simply doesn't apply. By all immediate evidence, both objective and subjective, the person still exists. The
only distinction is a discontinuity the the space-time vector of the body in question.
So, the question remains. Would you use the teleporter given the above facts or would you take the bus to work instead?
What facts? All I see is a definition.
As I've pointed out:
1. Everyone who uses the teleporter (as defined) will be entirely satisfied that it works as promised. There is not only no evidence of any other outcome, there is no possible evidence of any other outcome. You have defined an undetectible problem into existence arbitrarily.
2. Teleporters are impossible.