The authors define a subset \(S\) of a topological space \(X\) to be an \(\alpha^*\)-set if \(\text{int} S\) is regular open. A \(C\)-set is the intersection of an open set and an \(\alpha^*\)-set. A function between two spaces is called \(C\)-continuous if the inverse image of each open set is a \(C\)-set. Among some other observations the authors prove the following result: A function between topological spaces is continuous if and only if it is \(C\)-continuous and \(\alpha\)-continuous.