Concerning a characterization of continuity (Q1374614)

The author shows that a real-valued function \(f\) on a \(T_3\) locally connected Baire space is continuous if and only if (1) \(f\) is a Darboux function, (2) \(f\) is almost continuous in the sense of Husain, and (3) \(f\) is not of Cesàro type. This improves a characterization of continuity given by J. Smítal and E. Stanova. Moreover, it is proved that for a function \(f\), the following three conditions are not redundant: (1) \(f\) is extendable, (2) \(f\) is almost continuous in the sense of Husain, (3) \(f\) is not of Cesàro type.