If the Earth spun more slowly, that temperature differential would increase. If it was "stationary" - and by stationary, I mean tide-locked so that one side of the Earth always faced the sun, in the same way as the Moon is tide-locked with the Earth - the near side of the Earth would increase massively in temperature, while the far side of the Earth would plummet in temperature. It would certainly drop far enough to freeze water.
It might drop far enough to liquify or freeze air.
If it did, the far side of the Earth would be entirely uninhabitable: no atmosphere, and utterly freezing temperatures. Meanwhile, the near side of the Earth would be equally uninhabitable: still no atmosphere (it's all frozen on the far side!), and boiling temperatures.
Speed up the spin just a little, though, and you might get yourself into a situation where the only habitable spot on the planet was a narrow ring on the "edge" of the planet as seen from the Sun. The oceans would be entirely frozen, and the air would have thawed and resulted in atmosphere. I don't know if you could realistically have the air get cold enough to liquify, and still have a breathable atmosphere in some areas - the liquified air would, of course, occupy less volume, and the gaseous air would rush to fill that volume. But even if not, as long as the air doesn't get *quite* that cold, you could have this narrow habitable area.
Of course, everything on the surface would be dead, but as long as you carried your own farms along, you'd be fine. For certain definitions of "fine".
I'm imagining a universe in the far, far future. Humanity never got into space in any appreciable way, but never nuked themselves into oblivion either. It's so far in the future that the planet Earth is mere thousands of years away from tidelocking entirely and becoming uninhabitable . . . but for now, humanity survives on giant city-sized vehicles, slowly trundling across the frozen oceans and eroded mountains, now little more than hills, trying to stay within the breathable zone, trying to keep precisely in the area that's not too hot and not too cold.
In a few millenia they'll be doomed, of course. But humanity holds on.
I wonder what stories you could tell here.