In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire (1905). A Baire set is a set whose characteristic function is a Baire function (not necessarily of any particular class, as defined below).
Classification of Baire functions
- The Baire class 0 functions are the continuous functions.
- 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 n functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than n.
Some authors define the classes slightly differently, by removing all functions of class less than n from the functions of class n. 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
- 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 function equal to 1 if x is an integer and 0 otherwise. (An infinite number of large discontinuities.)
- The function that is 0 for irrational x 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 gives 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 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.
Baire class 2
- An example of a Baire class two 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. It is discontinuous everywhere.
Baire class 3
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- Baire, R. (1905), Leçons sur les fonctions discontinues, professées au collège de France, Gauthier-Villars
- Kechris, Alexander S. (1995), Classical Descriptive Set Theory, Springer-Verlag