  Given a strictly positive measure, we characterize inner semicontinuous solid
convex-valued mappings for which continuous functions which are selections
almost everywhere are selections. This class contains continuous mappings as
well as fully lower semicontinuous closed convex-valued mappings that arise in
variational analysis and optimization of integral functionals. The
characterization allows for extending existing results on convex conjugates of
integral functionals on continuous functions. We also give an application to
integral functionals on left continuous functions of bounded variation.
