Symmetric semicontinuity implies continuity

Abstract:

The main result of this paper shows that, for any function, symmetric semicontinuity on a measurable set $E$ implies continuity a.e. in $E$ and, similarly, that symmetric semicontinuity on a set residual in $R$ implies continuity on a set residual in $R$. These propositions are used to prove more precise versions of the fundamental connections between symmetric and ordinary differentiability.