Please provide a reference - though it's probably an average occupation. Like the average family having 2.6 kids - what is the significance of a fractional kid?
Yes. I know. Answer stands. By "reference" I mean like a book of an article you can show me that talks about the fractional occupation number, so I know the context and I'm not just guessing.
Fractional occupation is often used as a numerical integration technique for metals. If you have a Fermi surface and you're integrating over a discrete grid of points, the abruptness of the change in the occupation results in very slow convergence wrt the grid density. Using fractional occupations that smoothly go from 1 to 0 from the inside to the outside of the Fermi surface can be used to speed convergence.
There are different scenarios where fractional occupation numbers are useful. Here I give two: 1- They may indicate charge delocalization. So, a state which is supposed to be fully occupied is now fractionally occupied and the rest is spread over several ions. This situation could be the reality or an inherent error in DFT. 2- Fractionally occupied states my be intentionally simulated to check the energy dependence on the occupation number regarded as a continuous function. A correct description requires a vanishing second derivative for this function. Check figure 1 of this paper: http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.085202