In the ancient times the criteria for being Greek was "Greek by culture".
In the Byzantine times the word Greek had a religious meaning including the followers of paganism, an ancient religion. So in the middle ages we speak about "Greek by religion"

In the first period of the Ottoman era there was no such thing as "Greek". The determination "Greek" appears after the "New Greek enlightenment" at the end of 18th century (Elliniki Nomarhia, Rigas etc.). The criteria then was "Greek by ethnicity". Today's Greek state follows the racist law of "Greek by blood". A public declaration that you are Greek, the assistance of local state trustees who verify your healthy national beliefs, along with a document that one of your ancestors was recorded somewhere as Greek are the necessary conditions for one to qualify to be Greek. I have seen some funny discussions between ex-soviet citizens on how they would be recognized as Greeks. "My uncle was Greek, and you?" "The cousin of my aunt" was Greek, etc, etc.

The criteria of what is Greek changes from time to time throughout history.
Greek by culture
Greek by religion
Greek by ethnicity
Greek by blood
How can you claim continuity of a group of people throughout time since the criteria to define this group has changed 4 times in this course of history?