Comparison of observed (dashed) and derived (solid) XLFs at z = 0 (red), 1 (orange), 2 (green) and 3 (blue). The two are in relatively good agreement. Our simple assumption of a Gaussian spread in |$\dot{m}$| does not produce quite as sharp a ‘knee’ as in the observed XLF. Our simple model slightly overpredicts the XLF at z = 3.