For further information, see the 'Ship of Theseus paradox'.
Let's try a thought experiment:
Suppose an evil genius straps you to a table and starts removing parts of you. He cuts off your hand. Are you still you? Lots of people lose a hand and they usually claim to be the same person. The same might be said for arms and legs, sensory organs, and even important things like hearts, lungs, and so on.
That just leaves us with your brain. A brain-in-a-jar might think it was you, even without a body, and if our evil genius manufactured an artificial body that was just like your old one, we might not even notice much of a difference. But perhaps we could do the same with your brain. People kill brain cells with drugs and alcohol all the time and still say they are the same. Again, we might imagine that if every cell in your brain were replaced by our evil genius one at a time with an artificial duplicate, it might still be 'you', even though none of the original physical parts of you were there.
What is 'you', then, could be said to be the mind - the construct of the brain. A system instead of a physical thing. Something which might be duplicated on a computer, perhaps, or in some other way exist without a physical form of any kind. Not entirely inconceivable.
Now suppose our evil genius takes all those parts he took off of you and reassembles them in exactly the way he took them. The artificial you - which has had a continuous conception of being you - would be confronted by a new you made of the physical material of the old you. Yet if the new you's brain took up operating from where it left off, it might not know it wasn't you except for a few missing minutes of thought. Or put another way: if we introduced people to the two of you, how would they know which one was 'real'? It could be argued that when there is no measurable difference, there is no actual difference. And at no point did anything really die, per se.
Interestingly, we can take it in the opposite direction as well. You think of yourself as 'you' because you have a continuity of consciousness. But in the same wise, you don't think you are the same person as you were when you were six years old. You remember being that person, but that person is not who you are now. Even when the evil genius cuts off your hand, you are different. If he also cut off your memory of having a hand, you might be very distinctly so.
If we expand this out to all things which have an effect on who you think you are and how you behave, it ends up including almost everything. Instead of being just a mind, you become a conscious node of much of the local universe. Every change changes you to some extent, and every second in the past is another you which is gone and never likely to ever return. And again, because 'you' isn't necessarily some specific biological creature, 'you' may exist long after that creature is disassembled atom by atom.
It all depends on how you think about things. I don't think either answer is necessarily more correct than the other; rather they are different perspectives on the same thing. My two cents.