Edinburgh (the only city I really feel qualified to comment on) is a mix too. Some sides of the city have a sudden shift from urban to rural, some are more gradual.
Where I live - Leith - used to be a separate city from Edinburgh but now there's no obvious distinction.
EDIT: we also don't seem to talk about suburbs, even though there are named areas of the city. But I might have ideas wrong, since I've not spent any time in US cities so don't fully get what is meant by suburbs.
Also also: people often live in little villages outside of a major city and commute in, so I suppose that's kind of the same thing? They'll live there for schooling, or because houses in the city cost far more than you pay in an outlying village, or just because they don't want to live in a massive community.
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