Your unfortunate world is tidally locked... and had a relatively recent collision with another planet.
The planet orbits close enough to its star that it would normally be tidally locked, so in the absence of other factors, one side of the planet (which we'll call the south side) would constantly be facing the sun, and the other side (which we'll call the north side) would always face away. That gives you your eternal day and night at the poles.
For millions of years, it was tidally locked. Life sprang into being along its temperate equator, never knowing anything but eternal dusk, sandwiched between hemispheres of cold darkness and scorching daylight.
Then another planet, small compared to your world but still more massive than any asteroid ever to strike the Earth, collided with it. Since then, the momentum of the impact caused the planet to wobble. Not enough to fully break the tidal lock, but enough that a band near the equator spends half its time facing the sun and half its time facing away. This, effectively, gives you the day-night cycle you're looking for. The life that miraculously survived the impact has since recovered and has adapted well to the pattern of night and day, just like life on Earth has.
Eventually, tidal locking will slowly take over again, causing the wobble to fade, but this could take anywhere from a few million to hundreds of millions of years, depending on various parameters of the planet's orbit. In the meantime, this scenario fits your requirements to a T!