We prove that a countable regular space has a continuous selection if and only if it is scattered. Further we prove that a paracompact scattered space admits a continuous selection if each of its points has a countable pseudo-base. We also provide two examples to show that: (1) paracompactness can not be replaced by countable compactness even together with (collectionwise) normality, and (2) having countable pseudo-base at each of its points can not be omitted even in the class of regular Lindelöf linearly ordered spaces.
Seiji FUJII. Kazumi MIYAZAKI. Tsugunori NOGURA. "Vietoris continuous selections on scattered spaces." J. Math. Soc. Japan 54 (2) 273 - 281, April, 2002. https://doi.org/10.2969/jmsj/05420273