On analytic functions with cluster sets of finite linear measure (Q1090441)

Let f be a non-constant complex-valued function defined in the unit disc D. The total cluster set C(f) consists of all limits points of f(z) as \(| z| \to 1\), \(z\in D\). The linear Hausdorff measure of \(E\subset C\) (the complex plane) is denoted by \[ \Lambda (E)=\liminf_{\epsilon \to 0(D_ n)}\sum_{n}diam D_ n \] where the infimum is taken over all systems \((D_ n)\) of disks with diam \(D_ n<\epsilon\) that cover E. The author proves Theorem. If f is bounded and analytic in D and if \(\Lambda (C(f))<\infty\), then f has a continuous extension to \(\bar D.\) This result was proved by Globevnik and Stout [preprint, Ljubljana, 1985] under the additional assumption that \[ \iint_{D}| f'(z)|^ 2dxdy<\infty. \] The following comments are added in proof: H. Alexander [preprint 1986] has independently proved the same result. It has been recently generalized by J. J. Carmona and J. Cufí [preprint 1986].