  We introduce a new multi-power-law distribution for the Initial Mass Function
(IMF) to explore its potential properties. It follows on prior work that
introduced mechanisms accounting for mass accretion in star formation,
developed within the framework of general evolution equations for the mass
distribution of accreting and non-accreting (proto)stars. This paper uses the
same fundamental framework to demonstrate that the interplay between a
mass-dependent and a time-dependent step-like dropout rate from accretion leads
to IMFs that exhibit multiple power laws for an exponential mass growth. While
the mass-dependent accretion and its dropout is intrinsic to each star, the
time-dependent dropout might be tied to a specific history such as the rapid
consumption of nebular material by nearby stars or the sweeping away of some
material by shock waves. The time-dependent dropout folded into the
mass-dependent process of star formation is shown to have a significant
influence on the IMFs.
