  During the study of the topic of limit summability of functions (introduced
by the author in 2001), we encountered some types of functions that are related
to the mean value theorem. In this paper, we formally define mean value and
pointwise MV-functions associated with a given real function and extend some
aspects of the mean value theorem and properties. Also, we introduce and study
an induced conception which we call comparative convexity (and concavity). As
many applications of the study, we prove some uniqueness conditions for related
functional equations and completely solve several functional inequalities. It
is worth noting that there remain some important open problems and questions
for future studies of the topic.
