Abstract

Let \(D\) be a domain in \({\mathbb C}^n\), let \(f:D\to {\mathbb C}\) be a continuous function and denote its graph by \(\Gamma(f)=\{(z,f(z)\); \(z\in D\}\). \textit{N. Shcherbina} [Acta Math. 194, 203–216 (2005; Zbl 1114.32001)] proved that the graph is pluripolar if and only if \(f\) is holomorphic. The authors apply Shcherbina’s result to prove: If \(p\) is a non-constant holomorphic function on \({\mathbb C}^m\setminus S\), where \(S\) is a hypersurface in \({\mathbb C}^m\), and \(Z=\{(z,w)\in ({\mathbb C}\setminus S)\times D\); \(p(z)=f(w)\}\) is pluripolar, then \(f\) is holomorphic in \(D\). They also prove a variant of this theorem under the assumption that \(D\) is a product domain and the function \(f\) is separately continuous. Finally, they prove that if \({\mathbb B}^n\) denotes the unit ball in \({\mathbb C}^n\), \(f:{\mathbb B}^n\to {\mathbb C}\) is continuous on every line through \(0\) and in some neighourhood of \(0\), and \(\Gamma(f)\) is pluripolar, then there exists a set \(V\) which is a union of complex lines through \(0\) with \(\lambda_n(V)=0\) and a holomorphic function \(\tilde f\) on \({\mathbb B}^n\) such that \(\tilde f=f\) on \({\mathbb B}^n\setminus V\) and in some neighbourhood of \(0\).