What continuity exists?
A trust exists.
It was constituted at some moment in time, by some founding act, for some stewardship purpose. From that moment forward, the trust is a continuous thing. It does not begin again each morning. It does not need to be re-created when circumstances change.
Within the trust there is an office: the office of trustee. The office is older than its current holder and will outlast them. The office is the seat of stewardship. The holder is the person who, for now, sits in it.
This is the continuity that exists at the start of our story: the trust itself, and the office through which the trust is held.
The office is occupied by Trustee A.