at the high-mass end of the SMF.

In this paper we take advantage of the high-quality data from the ultra-violet to the mid-infrared (MIR) available through the NEWFIRM Medium-Band Survey (NMBS; @vandokkum09b) to derive more accurate measurements of the high-mass end of the SMF of galaxies at $3.0 \leq z < 4.0$ by significantly reducing the impact of random uncertainties and to characterize, for the first time, the observed and rest-frame properties of a mass-selected sample of galaxies at $3.0 \leq z < 4.0$. These results are made possible by the combination of accurate photometric redshifts and well-sampled spectral energy distributions (SEDs) of $K$-selected galaxies at $z>1.5$ delivered by the medium-band near-infrared (NIR) filters of the NMBS, as well as by its large surveyed area ($\sim$0.5 square degree).

This paper is structured as follows. In § \[sec-ss\] we present the mass-selected ($M_{\rm star}>2.5\times10^{11}$ M$_{\sun}$) sample used to measure the SMF of galaxies at $3.0 \leq z < 4.0$; in § \[sec-prop\] we present the observed and rest-frame properties of the galaxies in the mass-selected sample. The stellar mass function and densities of massive galaxies at $3.0 \leq z < 4.0$ are presented in § \[sec-mf\], while in § \[sec-dustold\] the 