Let U be a bounded
open set in the plane. We study the smoothness at boundary points of the
continuous functions on U which are analytic in U. A main result is the
characterization of sequences xn∈U, xn→ x, with the property that the functions
are of class Ck along xn at x. As an application of this characterization, we
can find an open set U for any twice continuously differentiable arc J such
that U contains J in its boundary and the functions are of class C∞ on
J.