On the sets of points of radial continuity of analytic functions (Q1375047)

Let \(D=\{z:|z|<1\}\subset\mathbb{C}\), \(\Gamma=\partial D\). A point \(\zeta\in\Gamma\) is called a point of radial continuity of a function \(f\) defined on \(D\), if there exists a limit \(\lim_{r\to1}f(r\zeta)\). The set of all points of radial continuity is denoted by \(H(f)\). Earlier separate necessary and sufficient conditions were known for a subset of \(\Gamma\) to be the set of the points of radial continuity of an analytic function in \(D\). The author characterizes such sets completely. Theorem. A set \(E\subset\Gamma\) is the set of all points of radial continuity of an analytic function in \(D\) if and only if \(E\) is a set of \(F_{\sigma\delta}\) type on \(\Gamma\) and moreover, for every arc \(\gamma\subset\Gamma\) either \(E\) is a set of the first category on \(\gamma\) or \(\gamma\) contains an arc \(\gamma'\) such that the set \(\gamma'\setminus E\) has the zero linear Lebesgue measure.