Continuity of memories as a proof of the existence of self Buddhism denies existence of any Self.But I think I can prove the existence of Self as fundamental by the logic of continuity of memories.
I define Self as someone who remembers the past essence.

Examples to demonstrate the applicability of the definition above:


Buddha. Buddha is the essence of Gautama. Buddha's knowledge is
immortal. Who remembers this essence? My answer will be that Self
remembers Buddha. And moreover because Buddha's knowledge is
immortal, that self becomes immortal and therefore fundamental.
MySelf. I exist. I can remember my past essence that Truth is
Supreme. Who remembers this essence of me ? My answer will be that
my Self remembers my essence.


Therefore we see that self can be found as someone who remembers past essence.This self is imperishable.
Which 'you'? The you that at 69 will likely become more conservative and less social? Or the you that at 5 had no concrete notion of Self, or the question the 'you' of about now poses? The me of 5 minutes ago had no knowledge nor memory of your question. There is a self - a self that is dependent on conditions - dependent on mother and father for birth, food for existence, conditions for death.
This is a conventional 'you' and such a notion is fine - convenient and useful within conditionality. The Buddha argued agaisnt an independent, permanent, unconditioned You - the soul or Atman, however envisaged.
Self is 'obliterated' through one link: of only those things which are observed directly by the senses (mind is included as a sense in the canon), all things are observed as subject to conditions for arising, subsisting and ceasing - subject to change.
Things that are subject to conditions, impermanent - as all that is observed is - cannot be the basis of a permanent and independent Self. (it's a tough switch to thinking.. for me for example, I kept 'summing up' all experience into one exact thing - but that is the process of making a permanent Self from an objective experience that only provides impermanent objects as building blocks!!). As such, if you think it, it's not Self.

Bodhidharma asked, “Can each of you say something to demonstrate your
understanding?”
Dao Fu stepped forward and said, “It is not bound by words and
phrases, nor is it separate from words and phrases. This is the
function of the Tao.” Bodhidharma: “You have attained my skin.”
The nun Zong Chi stepped up and said, “It is like a glorious glimpse of
the realm of Akshobhya Buddha . Seen once, it need not be seen again.”
Bodhidharma; “You have attained my flesh.”
Dao Yu said, “The four
elements are all empty. The five skandhas are without actual
existence. Not a single dharma can be grasped.” Bodhidharma: “You have
attained my bones.”
Finally, Huike came forth, bowed deeply in silence
and stood up straight. Bodhidharma said, “You have attained my
marrow.” (wiki)

Silence is as close to Self as you will get, as language is instantly limited by Gödel (and/or also being conditioned). Mahayana and Theravada Buddhism have sticking points, and whether the ultimate is seen as having a capital U gets a lot of chat, but at the very least, for the things that are the seen, the heard and the sensed, Self is at best an approximation of things as they are - subject to a frame of reference, relational, and dependently originated.