The paper analyzes the bivariate distribution of galaxies in the Sloan Digital Sky Survey (SDSS) data, focusing on the color-magnitude (CM) relations and luminosity functions of early- and late-type galaxies. The authors fit double-Gaussians to the color functions of galaxies separated by absolute magnitude bins, tracing the bimodality from luminous to faint galaxies. They find that the CM relations for both red and blue distributions can be described by a straight line plus a tanh function, with slopes of about \(-0.04\) and \(-0.05\). The transitions in galaxy properties occur around \(2 \times 10^{10} \mathcal{M}_\odot\) for the red distribution and around \(3 \times 10^{10} \mathcal{M}_\odot\) for the blue distribution. The red distribution has a shallower faint-end slope (\(\alpha = -0.8\)) compared to the blue distribution (\(\alpha \approx -1.3\)). The authors also discuss the implications of these findings for galaxy formation and evolution, suggesting that the red distribution may be formed through major galaxy mergers.The paper analyzes the bivariate distribution of galaxies in the Sloan Digital Sky Survey (SDSS) data, focusing on the color-magnitude (CM) relations and luminosity functions of early- and late-type galaxies. The authors fit double-Gaussians to the color functions of galaxies separated by absolute magnitude bins, tracing the bimodality from luminous to faint galaxies. They find that the CM relations for both red and blue distributions can be described by a straight line plus a tanh function, with slopes of about \(-0.04\) and \(-0.05\). The transitions in galaxy properties occur around \(2 \times 10^{10} \mathcal{M}_\odot\) for the red distribution and around \(3 \times 10^{10} \mathcal{M}_\odot\) for the blue distribution. The red distribution has a shallower faint-end slope (\(\alpha = -0.8\)) compared to the blue distribution (\(\alpha \approx -1.3\)). The authors also discuss the implications of these findings for galaxy formation and evolution, suggesting that the red distribution may be formed through major galaxy mergers.