I'm currently reading a novel where the story happens on a planet orbiting Proxima Centaury. As the planet is very close to the star, which is a red dwarf, it is tidally locked to the star like our Moon is to Earth. This means that one revolution around its axis is the same length as orbit around the star, making one side of the planet permanently face the star and the other side - permanently turned away from the star.
I got curious: is it possible for a planet to not revolve around its axis at all? How would day/year relations work? If I understand this correctly, the diurnal cycle would be the same length as a year, and both day and night would be half-year long, right?
Would such a planet be habitable?