Sets of points of symmetric continuity

We study the sets of symmetric continuity of real functions in connection with the sets of continuity. We prove that sets of reals of cardinality and subsets of weakly independent sets of reals are sets of symmetric continuity. The latter strengthens a similar result of Darji. We improve results of Fried and Belna saying that the set of points of symmetric continuity of a real function that are not continuity points does not contain a nonmeager set with Baire property and has inner measure zero by introducing another notion of smallness below meager and measure zero.