ABSTRACT: In an ongoing project of mapping the central region of our galaxy at three wavelengths in the Near Infrared, we obtained a K band ( lambda 2.2 micron ) mosaic of an area of DELTA alpha * DELTA delta ~ 650''*710'' (R_equiv ~ 15.8 pc for R0 = 8.5 kpc) centered approximately on Sgr A*. These observations were performed at the European Southern Observatory, La Silla, Chile, using the IRAC2B camera with the ESO-MPG 2.2m telescope yielding a detection limit of S_K' ~ 100 mu Jy and a completeness limit of S_\rm K' ~ 2000 mu Jy. An estimate of K band luminosities of different stars reddened by an absorption corresponding to a visual extinction of A_V ~ 31 mag shows that within these limits only MS stars earlier than spectral type B5, giants and supergiants will be observed as individual stars; they account for less than 10% of the stellar mass but for more than one half of the K band luminosity of the mosaic. MS stars later than B5 are not resolved and appear as an extended continuum in the mosaic. Special care was therefore taken to apply an observing technique which allows to recover the extended background emission. The resolved sources were separated from the unresolved background continuum by fitting Lorentzian distributions to the sources. The integrated characteristics, the radial dependence and the core radius of both the unresolved and resolved stars within the mosaic were determined. We also determine the K band luminosity functions (KLF's) of the ~ 6*104 resolved and fitted stars for both the mosaic and two subregions within the mosaic (Sgr A East and M-0.13-0.08) and discuss their relevance for an analysis of the stellar populations in Nuclear Bulge, Galactic Bulge and Disk. A model KLF based on the Salpeter IMF and an estimated total stellar mass of M_* ~ 1.2*108 Mo for the central 31.6 pc fits the observed K band continuum well and joins smoothly with the observed KLF for the more luminous stars.