What continuity exists?
A trust exists.
Within it, an office of trustee. Within the office, a holder — Trustee A — who occupies the office for now.
Somewhere inside the trust there is an inheritance rule, written in advance, describing what happens to the office under various terminating events. Somewhere outside the trust there is an authority competent, when the rule is unclear or when reasonable parties disagree about it, to interpret the rule on the trust’s behalf.
This is the continuity that exists at the start of our story: a trust, an office, a holder, a rule, and an authority. Each in its proper place.
The office is occupied by Trustee A.