The class $R$ and finely analytic functions

The paper is concerned with the class ${R}$ of holomorphic functions introduced by Gonchar, and the class $R^0$, which is a special case of the former. A holomorphic function $f$ in a neighbourhood of $0\in\mathbb{C}$ belongs to ${R}^0$, ${f}\in {R^0}$ if it admits rapid rational approximation in some closed ball $\overline{B}(0, r)$, $r > 0$. It is proved that in certain cases functions in the class $R$ are finely analytic in the whole of $\mathbb{C}$.
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