Symmetry considerations indicate that if the distributions for X and Y are U(−1, 1) and U(0, 1) then the PDF will be 0 outside [−1, 1] and −log(|x|)/2 inside. We get the same PDF when both distributions are U(−1, 1).
Suppose that PDFX is the PDF for X and that for some δ>0, (−1 ≤ x ≤ 1) → PDFX(x) > δ
and that Y is a sample from U(−1, 1)
then the PDF for XY will also have the singularity at 0, perhaps diminished by factor δ.
If PDFY has an ε like the δ for X then the distribution for XY will have a singularity at 0 diminished by a factor of δε.
Note that N(m, d) which is the normal distribution with mean m and standard deviation d is bounded away from 0 by some positive δ in the interval [−1, 1]. We reluctantly conclude that the PDF of the product of samples from any two normal distributions is unbounded near zero!
Curiously it seems not to matter much; δε is usually very very small.