Let's say you're on your deathbed and this is a great deal in the future. A doctor has offered you two choices. 1. You die and 2. Your memories, thoughts, and emotions can be downloaded on a machine so that you can continue to live. If you choose the second one...are you really still alive?
I'd have to say no, because, we are talking about one's consciousness. I think a new consciousness would begin, and the old "you" would die. This is because, when you look at it, it's not a "transfer" but a destructive copy process. All the attributes of your consciousness are first duplicated, then the original is destroyed. You would see the machine whir to life from your bed, and then the lights would go out.
On a similar note - I wouldn't step onto a star-trek transporter either. I am pretty sure I'd "die" and a doppelganger would take my place at the other location. The copy would be sure the transporter worked, unable to tell it's consciousness only began that second, and decide they are entirely safe and don't interfere with the continuity of a conscious being's experience, but that would be an illusion.
That.... or all continuity of consciousness is an illusion - for all I know the individual life-cycle of consciousness only exists for a millisecond, or until one falls asleep... no way to know for sure. Not being able to know - I think I'd play it safe.
Another question: Your friend who saved your life once is dying, and gets cloned and has his consciousness transferred. Before the original is destroyed, someone busts into the room and says they have a cure that will let him live in perfect health, but with a 10% of relapse. You, him, his clone all go rock climbing, and due to an accident, you can save the original or the clone. Which do you save, and does it matter?
Is the original destroyed? When I move software, I have two choices, copy it to a new location or simply move it to that location. If your thought patterns and other mental processes are moved rather than copied, you would still be you. Now, if the soul is a metaphysical bit that requires some sort of higher power to create or manipulate, you would no longer be you. If on the other hand, the soul is the aggregate of all of your experiences, the mental extension of your life, then you would still be you after being put in the machine.
(I don't think they can make a computer that is as complex as the brain or store a person in one, but that was not the focus of that question.)
About the Star Trek transporter.... I wouldn't get on one either, but going by the Canon facts, it's safe. You are broken down into your component parts and moved to a new location where you are reassembled according to the map made as you were disassembled. No different than moving a file in a computer.
I think continuity of consciousness is real, for a healthy human. Even when you are sleeping, you still hear and absorb input from your surroundings. Only when you lose contact with everything around you, IE. Blackouts, comas, extremely high fevers, do you lose that continuity. Perhaps that is why some people come out of experiences like that with severe personality changes.
In your climbing question, I would save my friend. He is known to me, and we have history. The clone is someone I just met and may or may not mean anything to me at this time. It does matter. The clone may remember everything my friend does, but since he awoke, he has been having experiences that make him different. He may be very close to my friend, but he is not the same. Plus, my friend has lived through a life threatening illness that I was there for, and that would change him even more. as well as bringing us closer together than the being that doesn't remember that.
Plus, my friend is the one who saved my life and I owe that person, not the fruit of his loins or whatever a clone would be considered.