On Boundary Properties of Conformal Mappings

The classes of \({{C}^{1}}\) -smooth domains bounded by a contour that is Lyapunov outside any neighborhood of its certain point such that the derivative of a conformal mapping onto the unit disk is continuous at this point are described. The description is given in terms of some spaces for a unit tangent vector on the boundary contour. Corresponding results for piecewise smooth domains are obtained as a consequence.