Continuity of memories as a proof of the existence of self

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.

You existed a moment before and the moment before that and the moment before that moment and so on.

No. The experience of life the moment before has vanished, forever. It did not "exist". It was only a "temporary existence". Therefore, it cannot be construed as "existence" or "non-existence". It is something that arises & ceases. Its an impermanent phenomena. Importantly, it was only an experience of life, that is, an experience of one to five aggregates. It was not a "Self". "Self" was only an idea imputed on that experience.

You can be sure that you were present before this moment because you remember your past.

Memory arises in the present. Most experiences cannot be remembered however. Also, what is remembered is not the same sense of 'self'. For example, the mind can think back to the past and think: "Was that really me in the past? How could I have been like that & how could I have done those things?". For example, since "I" become a Buddhist & experienced some mental transformations, I struggle to identity with things from the past. Today, I am a completely different person, to the point that the old 'self' has completely died. In other words, there are actions I did in the past which would be impossible for me to do today. This shows the idea of 'self' is a changing thing.

Note: This is unrelated to how the idea of 'self' can completely vanish in the mind; leaving only the experience of life. Life without self (which is peaceful).

Remembrance of past is the continuity, which is YOU.

No. It is not YOU. It is only the remembrance of life; the remembrance of the five aggregates. Please refer to the Khajjaniya Sutta, which states:

At Savatthi. “Bhikkhus, those ascetics and brahmins who recollect their manifold past abodes all recollect the five aggregates subject to clinging or a certain one among them. What five?

“When recollecting thus, bhikkhus: ‘I had such form in the past,’ it is just form that one recollects. When recollecting: ‘I had such a feeling in the past,’ it is just feeling that one recollects. When recollecting: ‘I had such a perception in the past,’ it is just perception that one recollects. When recollecting: ‘I had such volitional formations in the past,’ it is just volitional formations that one recollects. When recollecting: ‘I had such consciousness in the past,’ it is just consciousness that one recollects.

Therefore, bhikkhus, any kind of form whatsoever … Any kind of feeling whatsoever … Any kind of perception whatsoever … Any kind of volitional formations whatsoever … Any kind of consciousness whatsoever, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near, all consciousness should be seen as it really is with correct wisdom thus: ‘This is not mine, this I am not, this is not my self.’

This is called, bhikkhus, a noble disciple who dismantles and does not build up; who abandons and does not cling; who scatters and does not amass; who extinguishes and does not kindle.

You took birth , you loved your parents , events happened in your life

No. There is no idea or thought of "self" when a baby is born from its mother. The idea of "self" only arises when the mind matures & is able to start identifying with things. Modern psychology studies this. A mentally retarded or brain damaged child may struggle to develop a coherent idea of "self". "Self" is a thought. Please refer to the Mahāmāluṅkya Sutta, which states a new born child has no thoughts of self-identity:

For a young tender infant lying prone does not even have the notion ‘identity,’ so how could identity view arise in him?

all that contributed to your evolution as a Self.

This has some correctness. There is no inherent self but self is something that evolves. This shows self is not anything permanent or real but something that evolves & changes.

Being such, self can also devolve & disappear, such as in very old people who have dementia or in enlightened minds.

Rising of Self can be considered suffering

Very good. The Buddha taught the arising & conceiving of self is suffering (SN 12.15; SN 5.10) & a disease (Ud 3.10; MN 140). This is why it is not a self but is only suffering & only a disease. The Budddha said: "The world calls a disease 'self' " (Ud 3.10). Self is not a real thing but only a name, like calling a child 'John' or 'Mary'. There is nothing inherently real in the name John or Mary.

but one can not deny that you exist

No. 'Self' can be denied because it can be experienced to not exist. But life or the five aggregates cannot be denied. The original question is confusing 'self' with 'life'; confusing the 'name' with the 'thing'. Life is the thing. Self is the name. The name is not the thing. The name is only a name. Self is only a name given to a suffering disease. When the body of a child is hungry for food, the body cries. Later, when the child is older & can think, the mind develops the idea: "I am hungry". This shows self is the arising of suffering; is the arising of disease. 'Self' only arises when there is hunger & craving.

The rest of the question is just more non-sense & is produced by lust & fear. The question is a disease born from suffering.