  We explore using our continuity approach the underlying connections between
the evolution of the faint-end slope of the stellar mass function of
star-forming galaxies, the logarithmic slope of the sSFR-mass relation and the
merging of galaxies. We derive analytically the consequences of the observed
constancy of the faint-end slope since redshifts of at least z ~ 2. If the
logarithmic slope of the sSFR-mass relation is negative, then the faint-end
slope should quickly diverge due to the differential mass increase of galaxies
on the star-forming main sequence, and this will also quickly destroy the
Schechter form of the mass function. This problem can be solved by removing low
mass galaxies by merging them into more massive galaxies. We quantify this
process by introducing the specific merger mass rate (sMMR) as the specific
rate of mass added to a given galaxy through mergers. For a modest negative
value of the logarithmic slope of the sSFR-mass relation of beta ~ -0.1, an
average sMMR ~ 0.1sSFR across the population is required to keep the faint-end
slope constant with epoch, as observed. This in turn implies a merger rate of ~
0.2sSFR for major mergers, which is consistent with the available observational
estimates. More negative values of beta require higher sMMR and higher merger
rates, and the steepening of the mass function becomes impossible to control
for beta < ~ -0.6, for an observed value of the faint-end slope of ~ -1.4. The
close link that is required between the in situ sSFR and the sMMR probably
arises because both are closely linked to the buildup of dark matter haloes.
These new findings further develop the formalism for the evolving galaxy
population that we introduced earlier, and show how striking symmetries in the
galaxy population can emerge as the result of deep links between the physical
processes involved.
