Personal continuity in the absence of a persistent, unchanging self How is personal continuity (including continuity at rebirth) explained in Buddhism in the absence of a persistent, unchanging self? Do all Buddhists agree on a single explanation? I am reading the book, Foundations of Buddhism, by Rupert Gethin. He has provided some explanation, but I am not sure whether it is the original Buddhist explanation or his own interpretation.
The arising and cease of the aggregates (form, feelings, perception, mental formations and consciousness) occurs everytime contact happens. When contact occurs, new feelings, perceptions and consciousness arise, which are affected by the previous mind-states. As a result, mental formations are altered as well. And form is not the exception: with contact, multiple physiological changes happen at a molecular and cellular levels.

If we have a object, and we change or modify at least one of its contituents, we can no longer say that it is the same object anymore. Sure, some of its parts are still intact, but as a whole, the object is not the same. The same applies to the "self": if we took two point of time and compare that which we consider a "self" in both of those points, we can see that although it is not the same being, it is not an entirely different one, totally disconnected causally speaking.

The aggregates are influenced by external factors and stimuli, while at the same time they are influencing each other. Also, the khandhas are exerting a constant influence on the environment as well. There's constant change (what a paradoxical statement, isn't it?) in the aggregates and in the world. As nothing stays the same all the time, what deserves to be called "I" and "me" in a permanent and ultimate sense?

In the Anatta Lakkhana Sutta it is stated: 


  "Bhikkhus, how do you conceive it: is form permanent or impermanent?" — "Impermanent, venerable Sir." — "Now is what is impermanent painful or pleasant?" — "Painful, venerable Sir." — "Now is what is impermanent, what is painful since subject to change, fit to be regarded thus: 'This is mine, this is I, this is my self'"? — "No, venerable sir."


Also, when we act with intention (either through thoughts, speech or bodily actions) we create kamma. In the Kamma Sutta it is said: 


  "Now what, monks, is old kamma? The eye... The ear... The nose... The tongue... The body... The intellect is to be seen as old kamma, fabricated &amp; willed, capable of being felt. This is called old kamma. "And what is new kamma? Whatever kamma one does now with the body, with speech, or with the intellect: This is called new kamma."


As a consequence of that Kamma (through "our" volitional actions), the being "we were" created the being "we are" now, which in time it's creating the being "we will be" in the future. What "we are" now is the result of mutual conditionality between past kamma, present kamma and the present context/stimuli.

"We" always in quotations, because it has already been said that there's not a permanent thing to be called a "self" between all those states of being through time. And so, you cannot talk about "personal" continuity in the sense of a person which is carried on and underlying the changing aspect of the "self"; absolutely everything in the aggregates is subject to change if the supporting conditions cease to exist.

Kind regards!