  In the article a technique of the usage of $f$-continuous functions (on
mappings) and their families is developed. A proof of the Urysohn's Lemma for
mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension
Theorem for mappings is proven. Characterizations of the normality properties
of mappings are given and the notion of a perfect normality of a mapping is
introduced. It seems to be the most optimal in this approach.
