  Parfit could reply that continuity might not suffice for special care. It is not clear that F1 has reason to care specially about F3 — F3 might seem a stranger, perhaps even an unlikeable one.  When young, some worry about becoming someone they would not now like.  They see no reason for special care for this future person.  This worry makes sense, but if continuity were sufficient for special care, it would not.  If so, perhaps both continuity and connection, or perhaps continuity and admirability, are needed.  This would let Parfit keep the conclusion that I need not care specially for some of my future selves, but would not justify the conclusion that I have reason to care specially about other people who are merely connected to me now (or are merely admirable).