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We explore using our continuity approach the underlying connections betweenthe evolution of the faint-end slope of the stellar mass function ofstar-forming galaxies, the logarithmic slope of the sSFR-mass relation and themerging of galaxies. We derive analytically the consequences of the observedconstancy of the faint-end slope since redshifts of at least z ~ 2. If thelogarithmic slope of the sSFR-mass relation is negative, then the faint-endslope should quickly diverge due to the differential mass increase of galaxieson the star-forming main sequence, and this will also quickly destroy theSchechter form of the mass function. This problem can be solved by removing lowmass galaxies by merging them into more massive galaxies. We quantify thisprocess by introducing the specific merger mass rate (sMMR) as the specificrate of mass added to a given galaxy through mergers. For a modest negativevalue of the logarithmic slope of the sSFR-mass relation of beta ~ -0.1, anaverage sMMR ~ 0.1sSFR across the population is required to keep the faint-endslope constant with epoch, as observed. This in turn implies a merger rate of ~0.2sSFR for major mergers, which is consistent with the available observationalestimates. More negative values of beta require higher sMMR and higher mergerrates, and the steepening of the mass function becomes impossible to controlfor beta < ~ -0.6, for an observed value of the faint-end slope of ~ -1.4. Theclose link that is required between the in situ sSFR and the sMMR probablyarises because both are closely linked to the buildup of dark matter haloes.These new findings further develop the formalism for the evolving galaxypopulation that we introduced earlier, and show how striking symmetries in thegalaxy population can emerge as the result of deep links between the physicalprocesses involved.

Mass and Environment as Drivers of Galaxy Evolution III: The constancy of the faint-end slope and the merging of galaxies