I'm talking about the continuity of western civilization. Which clearly is continuous, from ancient Greece, to Rome and until today. The fact that it's not always a single empire doesn't really matter to a civilisation. Most civilisations are not civilisation states. And you shouldn't over interpret barbarian conquests. They don't usually replace the majority of the population, although they can change culture. Being overrun by barbarians who introduce poor governance is common enough in Chinese history anyway
Rome is simply an episode in western civilization. It's not roman civilisation. That's why you shouldn't compare Rome to China but to the Han dynasty
India is not culturally western. Turkey is not the inheritor of Rome either, despite ottoman claims. But people like the English have fully self colonised, abandoned their Celtic and Saxon heritage and are fully western, graeco Roman now. Their original culture by blood has only left a small legacy, but they wouldn't think of themselves as Celtic tribes, even if that's what the majority are genetically