Frankly, there's no way to prove consciousness remains continuous from moment to moment. There is no way to even internally verify the subjective experience of consciousness if it is actually not truly continuous, but reconstructs itself from state to state at a speed higher than we can perceive.
This is... interesting. Actually, I've only heard similar from Eliezer Yudkowsky IIRC, he believes that the Eliezer Yudkowsky of 5 minutes ago is about as much as the Eliezer Yudkowsky of now as an Eliezer Yudkowsky 30 meters away would be the Eliezer Yudkowsky sitting where he is... eh, it was
this video. Strange, when I think of it, since he also used continuity of consciousness as a plot point in Harry Potter and the Methods of Rationality. Then again, I might be interpreting his beliefs as the
exact opposite of what they actually are (it's either that consciousness is consistent over both time and space instead of neither, basically not treating time as a priveleged dimension, which is way Yudkowsky, what with all the timeless physics and decision theory and all, though I don't remember if he's changed his mind about all that).
I have no opinion. I don't think free will's real. That's about as far as I go with all that. I'm not sure trying to figure out the nature of consciousness is very scientific at all if there's no way to test it whatsoever. It's also difficult to test what with ethics and such. Gaps in time seem to allow consciousness to "resume", but what exactly does that mean? Is time priveleged in that way? Etc.