This is a complex and sometimes counterintuitive topic. Here is an alternative viewpoint.

There is no continuity of consciousness for human beings today. Every night when you go to sleep you experience dreamless periods during which you are not conscious.

Someone who strictly believed that continuity of consciousness was necessary to preserve identity would by unhappy every day when waking up. The “previous” person who went to sleep must be mourned because that person is “dead”. The person who wakes up is a “new” person.

I think very few people follow this kind of strict interpretation. We all know that there are discontinuities to consciousness and yet a sense of identity is preserved.