Personal identity, in my view, rests entirely on mental traits (memories, personality, habits, knowledge, etc.). The particular hardware which supports and implements these traits is irrelevant. Under this assumption, gradual replacement is no different from discontinuous replacement. Let the surgeon replace my neurons one at a time, or all at once. Even let my brain be taken apart completely, wait a week, and then reconstruct it with different hardware. In any case, the resulting brain is functionally equivalent to the one before the operation -- and this is all that matters. The patient survives, the same person before and after.

(Note, too, that if the surgeon is imperfect, and fails to replicate the original brain completely, we are not befuddled -- we merely say that the patient mostly survives, and is mostly the same person afterwards. We would describe most current-day brain surgery in the same terms, for that matter.)