Symmetric functions whose set of points of discontinuity is uncountable (Q1100295)

A real function f of a real variable is said to be symmetric (symmetrically continuous) if \[ \lim_{h\to 0}(f(x+h)+f(x-h)- 2f(x))=0\quad (\lim_{h\to 0}(f(x+h)-f(x-h))=0) \] is fulfilled for every x. The author constructs a measurable symmetric function f, whose set of points of discontinuity is uncountable. This gives an affirmative answer to a question raised by \textit{L. Larson} [Real Anal. Exch. 9, 154- 178 (1984; Zbl 0574.26005)]. Furthermore, the function \(g=| f|\) is an example of a symmetrically continuous function discontinuous on an uncountable set.