Then who is a person with alzheimer's disorder. they do not concretise new memories. So are they the 'you' that existed with thier last new memory?
I ask because it is a salient point, Most people view memory as a crucial part of self consciousness, but even without alzheimer's memory is a tricky little bug that plays tricks on you.
It is a salient point, and memory is indeed a tricky little bug.
What this means is not that there is an identifiable instantaneous self, but that consciousness is also a tricky little bug.
The memory is not really storage like a hard drive, it is more a reconstruction of the past from limited data sets. And it is prone to error, change and confabulation.
Where does you go during anesthesia or after a stroke?
Much the same place it goes during sleep. Brain processing is suppressed, and conscious awareness is suppressed with it. Memory remains intact (essentially), and when we awake and processing starts up again, consciousness also starts up again. Like the sleep button on a laptop; you don't lose your place, but nothing happens while you're hibernating.
That's simplifying things a lot, of course; sleep isn't all the same and it's also distinct from other forms of unconsciousness.
I am open to change but I have considered this a long time.
We're agreeing on far more than we're disagreeing, so it might just be the way we've chosen to express our respective points. What I take rocketdodger as saying, and this is my view as well, is that when brain processing is suspended there is no you as such. We still refer to you as you only because we have an expectation (reasonable or not) that processing will resume.
So when you say continuity is an illusion, well, I'd say it is and it isn't.

We expect our - and others' - perception of selfhood to change over time; that change is something we consider an attribute of consciousness. So with all these discontinuities going on - sleep, neurological change, mental change - we look for hints of continuities. Physical continuity - I look the same as I did yesterday, so it's still me. Mental continuity - yesterday I liked chocolate; today I like chocolate, so it's still me. It gets tricky when you start to play with thought experiments like replacing one neuron at a time, or worse, start using the new hi-tech neurons to duplicate the mind in another substrate.