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