Baire function

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).

Baire functions of class n, for any countable ordinal number n, form a vector space of real-valued functions defined on a topological space, as follows.

• 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.

For example, the derivative of any differentiable function is of class 1.

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.

• 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, $\chi _{\mathbb {Q} }$, also known as the Dirichlet function. It is discontinuous everywhere.

• 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.

See also

• Baire set
• Nowhere continuous function

References

• Baire, R. (1905), Leçons sur les fonctions discontinues, professées au collège de France, Gauthier-Villars

External links

• Springer Encyclopaedia of Mathematics article on Baire classes