On Fatou's theorem

Let E be a set on the unit circle T of C. We prove that there exists an f is an element of H-infinity which has no radial limits on E but has unrestricted limit at each point of T\E if and only if E is an F-sigma of measure zero. The necessity of the condition that E is an F-sigma is almost obvious and the necessity of the condition that E is of measure zero follows from Fatou's theorem.