The notion that everyone is constantly changing is in no way surprising. However, the implications of this are often not considered. An interesting corollary of it is that legitimate immortality is not possible, even if humans could live forever. This would merely be an infinite series of births and deaths of different individuals. In light of this, fear of death seems foolish. We do not conventionally fear change in ourselves which, as I have described, results in death of the individual. The only way in which physical death is different is that it ends the succession of new related individuals. This is not something which should trouble your present self, as you would have died either way.
A final item worth addressing is how the notion of the soul would affect what I have suggested. The soul could be suggested as a means by which the continuous death I describe could be averted. If the soul exists, is eternal, and unchanging then I must admit that death is an impossible concept if the soul is defined as the individual. However, the very idea of the soul is that it is the essence of a person. Surely then the soul relates to the personality, memories and/or the conscious experience of the individual (I argued above that conscious experience is itself affected by personality and memories). If the soul is constant then it must not be related to these qualities, because they are known to change. This sort of soul would therefore not represent the essence of the individual in any meaningful way. Hence, the soul must vary over time and thus is subject to the same continuous death as all other aspects of the individual.
 Josefsson K, Jokela M, Cloninger CR, Hintsanen M, Salo J, Hintsa T, Pulkki-Raback L, Keltikangas-Jarvinen L. 2013. Maturity and change in personality: Developmental trends of temperament and character in adulthood. Dev Psychopathol. 25 (3): 713-27.
 Chua EF, Hannula DE, Ranganath C. 2012. Distinguishing highly conﬁdent accurate and inaccurate memory: Insights about relevant and irrelevant inﬂuences on memory conﬁdence. Memory. 20 (1): 48-62.