In this paper the authors follow a path that has become popular recently: to introduce and investigate a notion of abstract subdifferential for lower semi-continuous functions in Banach spaces by requiring the candidate for subdifferential to satisfy a priori given (inspired from subdifferential calculus) properties. Using an appropriate technical property the authors show that for each such subdifferential a mean-value theorem is true. Further, a class of functions generalizing the convex ones is investigated and naturally related to the notion of \(\phi\)-monotonicity of certain of their subdifferentials. Finally, the results obtained and the techniques developed are used to study the integrability of lower semi-continuous functions.