The effect of compressibility and rarefaction on the development of temporally developing mixing layers through numerical simulation employing the unified gas kinetic sche me (UGKS) is investigated in this work. In the low Knudsen number range, mixing layer undergoes Kelvin-Helmholtz instability (KHI), which is characterized by shear layer roll up about the inflection line. The combined influence of Mach number and Knudsen number on KHI points to the contrasting mechanisms of compressibility and rarefaction. Five different regimes in the Reynolds-Mach-Knudsen number range are identified. The action of compressibility is through pressure fluctuations, and that of rarefaction is through viscous action. Further increase in Knudsen number, leads to breakdown in continuum hypothesis, which is observed as deviation from Navier-Stokes-Fourier formulation. Three modes of deviation from continuum are identified using the Grad-13 moment equation. The parameters scaling these deviations in each of these regimes are different. It is found that the components of stress tensor and heat flux vector do not scale in a similar manner when Knudsen number and Mach number are varied.