We study the spherically symmetric collapse of a perfect fluid using area-radial coordinates. Recently, Giambò et al. derived a second-order quasi-linear partial differential equation for the mass function. They claimed that singularities formed in the collapse are naked if the mass function is analytic with respect to the area-radial coordinates in the neighborhood of the singularities. We point out limitations of the analytic mass functions, propose new expressions for the mass function and investigate the relationship between those and the nature of the singularities. © 2006 American Institute of Physics.