People get way too worked up about this.
Be less "Guy Fleegman afraid he was a redshirt" and more "Guy Fleegman once he's realized he's comic relief".

If a consciousness thinks it's continuous that consciousness is continuous.
The substrate your consciousness dances on also changes all the time. Molecules arranged around the galaxy or cells dying and being replaced pose the exact same quandary, and the solution to both would seem to be "who cares"?

The arrangement of cells and neurons known as "You" goes in, the arrangement of cells and neurons known as "You" comes out.