Baire functions of class α, for any countable ordinal number α, form a vector space of real-valued functions defined on a topological space, as follows.
This means that each Baire function has a well defined class, but the functions of given class no longer form a vector space.
Henri Lebesgue proved that (for functions on the unit interval) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class.
Examples: The Baire Characterisation Theorem states that a real valued function f defined on a Banach space X is a Baire-1 function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuity relative to the topology of K. By another theorem of Baire, for every Baire-1 function the points of continuity are a comeager Gδ set (Kechris 1995, Theorem (24.14)).
An example of a Baire class 2 function on the interval [0,1] that is not of class 1 is the characteristic function of the rational numbers,