  Multilevel Splitting methods, also called Sequential Monte-Carlo or
\emph{Subset Simulation}, are widely used methods for estimating extreme
probabilities of the form $P[S(\mathbf{U}) > q]$ where $S$ is a deterministic
real-valued function and $\mathbf{U}$ can be a random finite- or
infinite-dimensional vector. Very often, $X := S(\mathbf{U})$ is supposed to be
a continuous random variable and a lot of theoretical results on the
statistical behaviour of the estimator are now derived with this hypothesis.
However, as soon as some threshold effect appears in $S$ and/or $\mathbf{U}$ is
discrete or mixed discrete/continuous this assumption does not hold any more
and the estimator is not consistent.
  In this paper, we study the impact of discontinuities in the \emph{cdf} of
$X$ and present three unbiased \emph{corrected} estimators to handle them.
These estimators do not require to know in advance if $X$ is actually
discontinuous or not and become all equal if $X$ is continuous. Especially, one
of them has the same statistical properties in any case. Efficiency is shown on
a 2-D diffusive process as well as on the \emph{Boolean SATisfiability problem}
(SAT).
