  Preintegration is a technique for high-dimensional integration over
$d$-dimensional Euclidean space, which is designed to reduce an integral whose
integrand contains kinks or jumps to a $(d-1)$-dimensional integral of a smooth
function. The resulting smoothness allows efficient evaluation of the
$(d-1)$-dimensional integral by a Quasi-Monte Carlo or Sparse Grid method. The
technique is similar to conditional sampling in statistical contexts, but the
intention is different: in conditional sampling the aim is to reduce the
variance, rather than to achieve smoothness. Preintegration involves an initial
integration with respect to one well chosen real-valued variable. Griebel, Kuo,
Sloan [Math. Comp. 82 (2013), 383--400] and Griewank, Kuo, Le\"ovey, Sloan [J.
Comput. Appl. Maths. 344 (2018), 259--274] showed that the resulting
$(d-1)$-dimensional integrand is indeed smooth under appropriate conditions,
including a key assumption -- the integrand of the smooth function underlying
the kink or jump is strictly monotone with respect to the chosen special
variable when all other variables are held fixed. The question addressed in
this paper is whether this monotonicity property with respect to one well
chosen variable is necessary. We show here that the answer is essentially yes,
in the sense that without this property the resulting $(d-1)$-dimensional
integrand is generally not smooth, having square-root or other singularities.
