In [6], a sufficient condition for a space to be δ-stratifiable was presented and it was asked whether the condition is necessary. Jin et al [4] gave a negative answer to the question by showing that a space with the condition is zero-dimensional. In this paper, we show that a space with the condition is precisely an almost discrete space. Moreover, we introduce the notions of strongly lower (upper) semi-continuous functions, with which the characterizations of δ-stratifiable spaces are presented.