  We study the mappings of the first resolvable class defined by G. Koumoullis
as a valuable tool to address the point of continuity property in the
non-metrizable setting. First, we investigate the distance of a general mapping
to the family of mappings of the first resolvable class via the
\emph{fragmentability} quantity. We partially generalize papers of B. Cascales,
W. Marciszewski, M. Raja; C. Angosto, B. Cascales, I. Namioka; and J.
Spurn\'{y}. Second, we introduce the class of mappings with the countable
oscillation rank, study its basic properties and relate it to the mappings of
the first resolvable class and other well known classes of mappings. This rank
has been in a less general context considered by S.~A. Argyros, R. Haydon and
some others.
