  We study singularity formation for the pressureless Euler-Poisson system of
cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when
its smooth solutions experience $C^1$ blow-up, the $L^\infty$ norm of the
density becomes unbounded, which is often referred to as a delta-shock. We
provide a constructive proof of singularity formation to obtain an exact
blow-up profile and the detailed asymptotic behavior of the solutions near the
blow-up point in both time and space. Our result indicates that at the blow-up
time $t=T_\ast$, the density function is unbounded but is locally integrable
with the profile of $\rho(x,T_\ast) \sim (x-x_*)^{-2/3}$ near the blow-up point
$x=x_\ast$. This profile is not yet a Dirac measure. On the other hand, the
velocity function has $C^{1/3}$ regularity at the blow-up point. Loosely
following our analysis, we also obtain an exact blow-up profile for the
pressureless Euler equations.
