  We use a high resolution $\Lambda$CDM numerical simulation to calculate the
mass function of dark matter haloes down to the scale of dwarf galaxies, back
to a redshift of fifteen, in a 50 $h^{-1}$Mpc volume containing 80 million
particles. Our low redshift results allow us to probe low $\sigma$ density
fluctuations significantly beyond the range of previous cosmological
simulations. The Sheth and Tormen mass function provides an excellent match to
all of our data except for redshifts of ten and higher, where it overpredicts
halo numbers increasingly with redshift, reaching roughly 50 percent for the
$10^{10}-10^{11} \msun$ haloes sampled at redshift 15. Our results confirm
previous findings that the simulated halo mass function can be described solely
by the variance of the mass distribution, and thus has no explicit redshift
dependence. We provide an empirical fit to our data that corrects for the
overprediction of extremely rare objects by the Sheth and Tormen mass function.
This overprediction has implications for studies that use the number densities
of similarly rare objects as cosmological probes. For example, the number
density of high redshift (z $\simeq$ 6) QSOs, which are thought to be hosted by
haloes at 5$\sigma$ peaks in the fluctuation field, are likely to be
overpredicted by at least a factor of 50%. We test the sensitivity of our
results to force accuracy, starting redshift, and halo finding algorithm.
