  A class of radial, polynomial cutoff functions $f_{\textrm{c}n}(r)$ for
short-ranged pair potentials or related expressions is proposed. Their
derivatives up to order $n$ and $n+1$ vanish at the outer cutoff $r_\textrm{c}$
and an inner radius $r_\textrm{i}$, respectively. Moreover, $f_{\textrm{c}n}(r
\le r_\textrm{i}) = 1$ and $f_{\textrm{c}n}(r\ge r_\textrm{c})=0$. It is shown
that the used order $n$ can qualitatively affect results: stress and bulk
moduli of ideal crystals are unavoidably discontinuous with density for $n=0$
and $n=1$, respectively. Systematic errors on energies and computing times
decrease by approximately 25\% for Lennard-Jones with $n=2$ or $n=3$ compared
to standard cutting procedures. Another cutoff function turns out beneficial to
compute Coulomb interactions using the Wolf summation, which is shown to not
properly converge when local charge neutrality is obeyed only in a stochastic
sense. However, for all investigated homogeneous systems with thermal noise
(ionic crystals and liquids), the modified Wolf summation, despite being
infinitely differentiable at $r_\textrm{c}$, converges similarly quickly as the
original summation. Finally, it is discussed how to reduce the computational
cost of numerically exact Monte Carlo simulations using the Wolf summation even
when it does not properly converge.
