it's me because it's continuous

Consciousness persists because of the constant flow of information from our body to our brain. The continuity of the input leads to the continuity of consciousness. Mirror neurons allow us to imagine how others feel, for a short period of time. Do mirror neurons also allow us to imagine how our own body feels? Does our brain simulate what it would feel like to be us? It's the continuity of the "me" simulation that allows us to know it's "me." Breathing, heartbeat, etc.

Compare Proust (III, 645) where his love for Albertine dies but not his love of his own life. He compares this to an old mistress we can't get rid of.