The Baire class 1 functions are those functions which are the
pointwise limit of a
sequence of Baire class 0 functions.
In general, the Baire class α functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than α.
Some authors define the classes slightly differently, by removing all functions of class less than α from the functions of class α. 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.
Baire class 1
Examples:
The
derivative of any
differentiable function is of class 1. An example of a differentiable function whose derivative is not continuous (at x = 0) is the function equal to when x ≠ 0, and 0 when x = 0. An infinite sum of similar functions (scaled and displaced by
rational numbers) can even give a differentiable function whose derivative is discontinuous on a dense set. However, it necessarily has points of continuity, which follows easily from The Baire Characterisation Theorem (below; take K = X = R).
The characteristic function of the set of
integers, which equals 1 if x is an integer and 0 otherwise. (An infinite number of large discontinuities.)
Thomae's function, which is 0 for
irrationalx and 1/q for a rational number p/q (in reduced form). (A dense set of discontinuities, namely the set of rational numbers.)
The characteristic function of the
Cantor set, which equals 1 if x is in the Cantor set and 0 otherwise. This function is 0 for an uncountable set of x values, and 1 for an uncountable set. It is discontinuous wherever it equals 1 and continuous wherever it equals 0. It is approximated by the continuous functions , where is the distance of x from the nearest point in the Cantor set.
The Baire Characterisation Theorem states that a real valued function f defined on a
Banach spaceX is a Baire-1 function if and only if for every
non-emptyclosed 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
comeagerGδ set (
Kechris 1995, Theorem (24.14)).
Baire class 2
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, , also known as the
Dirichlet function which is
discontinuous everywhere.
Proof
We present two proofs.
This can be seen by noting that for any finite collection of rationals, the characteristic function for this set is Baire 1: namely the function converges identically to the characteristic function of , where is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over , where is an enumeration of the rationals. It is not Baire-1 by the theorem mentioned above: the set of discontinuities is the entire interval (certainly, the set of points of continuity is not comeager).
The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
^T. Jech, "
The Brave New World of Determinacy" (PDF download). Bulletin of the American Mathematical Society, vol. 5, number 3, November 1981 (pp.339--349).