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Baire* 1 functions from [0,1] to R were defined by R. J. O'Malley. For a general topological space X, a function f: X R will be said to be Baire* 1 if and only if for every nonempty closed subset H of X, there is an open set U such that U n H * 0 and f Il H is continuous on U. Several characterizations of Baire* 1 functions are found by altering the well-known Baire 1 characterization: If H is a nonempty closed subset of the domain of f, then f I H has a point where f I H is continuous. These conditions simply replace "closed subset of the preceding characterization with "subset ", "countable subset" or " dense-in-itself subset ". The relationships of these characterizations are examined with the domain of f being various spaces. The independence of these conditions from the discrete convergence condition described by A. Csaszair and M. Laczkovich is discussed.

Characterizations of Baire*1 functions in general settings