We consider a modified Press-Schechter (PS) formalism to derive an extended mass function for primordial black holes (PBHs), considering their formation by the collapse of energy density fluctuations. Said fluctuations are assumed to obey Gaussian statistics and they are obtained from a primordial power spectrum of broken power-law form with a blue spectral index for small scales. We also develop a method for converting existing constraints on the PBH mass fraction (relative to the total dark matter), derived assuming monochromatic mass distributions for PBHs, into constraints applicable for extended PBH mass functions. This new method takes into account the fact that, for any given constraining physical process (e.g., Big Bang nucleosynthesis or microlensing), the PBH masses that are constrained correspond to a finite range of the entire possible masses, and thus they apply to only a fraction of the total mass encoded in an extended mass function. Furthermore, the method also accounts for the redshift evolution of the mass function by, for example, black hole evaporation. We use our method for placing constraints on the parameter space of our newly-derived PS mass function, and we find that there are regions in parameter space where all the dark matter can be made of PBHs. Of special interest is the region for M * of order 10 2 solar masses, for a wide range of blue spectral indices, as this is close to the black hole masses detected by LIGO which are difficult to explain by stellar collapse.