  In this paper, it shows that for each fuzzy set $u$ on $\mathbb{R}^m$, the
set $D(u)$ is at most countable. Based on this, it modifies the proof of
assertion (I) in step 2 of the sufficiency part of Theorem 4.1 in paper:
Characterizations of compact sets in fuzzy sets spaces with $L_p$ metric,
http://arxiv.org/abs/1509.00447.
