  In this paper we posit that galaxy luminosity functions (LFs) come in two
fundamentally different types depending on whether the luminosity traces galaxy
stellar mass or its current star formation rate (SFR). Mass function types
reflect the older stars and therefore the stellar mass distribution, while SFR
function types arise from the young stars and hence the distribution of SFRs.
Optical and near-infrared LFs are of the mass function type, and are well fit
by a Schechter function (power law with an exponential cutoff at the bright
end). In contrast, LFs of the SFR function type are of a different form, one
that cannot be adequately described by a Schechter function. We demonstrate
this difference by generating SFR distributions for mock samples of galaxies
drawn from a Schechter stellar mass distribution along with established
empirical relations between the SFR and stellar mass. Compared with the
Schechter function, SFR distributions have a shallower decline at the bright
end, which can be traced to the large intrinsic scatter of SFRs at any given
stellar mass. A superior description of SFR distributions is given by the
"Saunders" function, which combines a power law with a Gaussian at the high
end. We show that the Schechter-like appearance of UV and H alpha LFs, although
they are LFs of SFR function type, results when luminosities are not corrected
for dust, or when average statistical corrections are used because individual
attenuation measurements are not available. We thus infer that the
non-Schechter form of the far-IR LFs is a true reflection of the underlying SFR
distribution, rather than the purported artifact of AGN contamination.
