I disagree. Your individual consciousness will diverge of course (you are just a meat machine) but your computer copy will continue on and your ideas, ideals, and personality would continue.

Writing this I just had a major existential crisis because I realized that at the exact point of divergence there is a point where 'you' are both alive and dead at the same time, Schrodinger's Tomb.

I agree with you in that I think a gradual change is less crisis inducing even though it equals out to the same thing. I think that I'd get a brain backup anyway, just in case.