This article concerns the Gleason property as a local phenomenon. We prove that there always exists an open set where the domain D (sic) C(2) has the Gleason beta property whenever the boundary of the Nebenhulle of D coincides with a C(2) smooth part of the boundary bD; here beta is either one of the Banach algebras, H(infinity) or A. As an easy consequence of this, we see that if the extremal boundary points are C(2)-smooth, then D has the Gleason beta property close to those points. Also a partial derivative-problem for locally supported forms is solved.