Ultrathin Cu films with thicknesses d between similar to 10 and 40 nm were thermally evaporated onto similar to 500-nm thick SiO2 on Si(100) substrates in an ultra high vacuum (UHV) chamber with a base pressure of 5 x 10(-10) torr. The sheet resistance R (Omega/square), was measured in situ at different film thicknesses by a collinear four-point probe. The infinite R at d < 10 nm suggested that the film consisted of discontinuous islands at these thicknesses. The R-value dropped rapidly by more than an order of magnitude when the thickness was increased from 10 to 15 nm, indicating the coalescence of islands. Further increases in d resulted in the R value gradually leveling off at 0.65 +/- 0.01 /square, corresponding to a resistivity rho of 2.67 mu Omega cm at d similar to 41 nm. The rho -d data were fitted by models that assume surface, interface and grain boundary scattering to be dominant mechanisms for the thickness dependence on resistivity. Models that do not include surface roughness do not fit our data in the sub-15-nm thick Cu films regime. The surface roughness was measured by atomic force microscopy (AFM). Our analysis shows that Namba's model that uses the measured surface roughness provides the best description of the resistivity-thickness behavior in sub-40-nm thick Cu films.