  We present a new analysis of stellar mass functions (MF) in the COSMOS field
to fainter limits than has been previously probed to z~1. Neither the total nor
the passive or star-forming MF can be well fit with a single Schechter function
once one probes below 3e9 Msun. We observe a dip or plateau at masses ~1e10
Msun, and an upturn towards a steep faint-end slope of -1.7 at lower mass at
any z<1. This bimodal nature of the MF is not solely a result of the blue/red
dichotomy. The blue MF is by itself bimodal at z~1. This suggests a new
dichotomy in galaxy formation that predates the appearance of the red sequence.
We propose two interpretations for this bimodality. If the gas fraction
increases towards lower mass, galaxies with M_baryon~1e10 Msun would shift to
lower stellar masses, creating the observed dip. This would indicate a change
in star formation efficiency, perhaps linked to supernovae feedback becoming
much more efficient. Therefore, we investigate whether the dip is present in
the baryonic (stars+gas) MF. Alternatively, the dip could be created by an
enhancement of the galaxy assembly rate at ~1e11 Msun, a phenomenon that
naturally arises if the baryon fraction peaks at M_halo ~1e12 Msun. In this
scenario, galaxies occupying the bump around M* would be identified with
central galaxies and the second fainter component having a steep faint-end
slope with satellites. While the dip is apparent in the total MF at any z, it
appears to shift from the blue to red population, likely as a result of
transforming high-mass blue galaxies into red ones. At the same time, we detect
a drastic upturn in the number of low-mass red galaxies. Their increase with
time reflects a decrease in the number of blue systems and so we tentatively
associate them with satellite dwarf galaxies that have undergone quenching.
