Just last night I thought about this and went to sleep in despair :>
If someone scans my brain, builds an artificial copy, instantly cuts out my brain and inserts the new one, I clearly die (because I got my brain/me cut out) and my copy lives on. From the outside, I'm the same, but the first me died.
So instead, I get my neurons replaced with artificial ones, one at a time. If I drink a few beers, brain cells die but I'm still conscious. So gradually replacing my neurons one by one should be no problem; there clearly is no loss of continuity, I'm still me.
But what's the difference? Let's assume one neuron is replaced every second. No problem, nice and gradual. What if it's one every millisecond, every picosecond? Should make no difference, because it's still one by one. But, the faster you do it, the more it approaches "instant". And instantly replacing all my brain cells is the same as replacing my whole brain. Thus, I'm dead and my copy lives on...