Separately Continuous Functions with Closed Graphs

Abstract

In this paper we prove that if $f:\R\times\R\to\R$ has a closed graph and all of its $x$-sections are continuous, and at least one $y$-section is continuous, then $f$ is continuous. It was already proved by Piotrowski and Wingler [PW] that if $f:\R\times\R\to\R$ has a closed graph and is separately continuous, then $f$ is continuous. Our result is stronger.