On controlled extensions of Carathéorody functions (Q2371908)

This is a technical paper dealing with Carathéodory and \(R\)-Carathéodory functions [\textit{A. Kucia}, J. Math. Anal. Appl. 223, No.~1, 302--318 (1998; Zbl 0981.28009)], and \((E,u)\)-controlled extensions [\textit{J. Melesic, D. Repovs} and \textit{P. Semenov}, J. Math. Anal. Appl. 285, No. 1, 62--73 (2003; Zbl 1034.54012)]. It is proved (Theorem 2) that, under some natural assumptions, the existence of \((E,u)\)-controlled continuous extensions implies the existence of \((E,u)\)-controlled extensions. In the proof, a technique using the Marczewski function is used.