I think continuity is a core component of self, and here's a quick argument why. Suppose someone were to come to you and tell you that they'd just created you, whole hog, the day before. They show you how they were able to do this and convince you that it's true. You have a memory of having done something bad a week ago, and upon investigating, you find that another 'you', created the same way, had actually done the bad act. Would you still feel guilty for having done the bad act? I think not, because in some necessary sense, that other 'you' isn't the you that you are today. (It's very confusing. But I think we have an intuition that continuity is a necessary feature of self. I think this is because in some sense we 'own' our past actions. They belong to us. If they weren't the product of our current brain and body, we wouldn't feel they belong to us. And a large part of what determines who we are is our past actions. If I hadn't lived the life I had lived, I wouldn't be the same 'me'.)
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