I never said that. I said Susa/Sush, the beginning of Iranian civilization, is nearly as old as Sumer, by about 500 years. Historians generally consider Sumer the oldest true civilization, even if there are older settlements that didn't rise to the level of complexity rating as civilization.

Being occupied by foreign powers doesn't break cultural continuity, otherwise you'd claim the modern Greeks don't have continuity with ancient Athens, despite speaking the same language.

Continuity also doesn't mean being static. Modern Greek is not the same as classical Attic. Modern Greeks are Christian, not again, and so on.