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.