Baire function

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In mathematics, Baire functions refer to certain sets of functions. They are studied in several fields of mathematics, including real analysis and topology.

Baire functions of class n, for any ordinal number n, are a set of real-valued functions defined on the real line, 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, but are not of Baire class 0.

In general, the Baire class n functions are all functions which are the pointwise limit of a sequence of functions each of which has Baire class less than n, but do not themselves appear in any lower-numbered class.

Many important functions in analysis which are not continuous are of Baire class one. For instance, the derivative of any differentiable function is either continuous (class 0) or class 1.

Henri Lebesgue proved that each Baire class is non-empty, 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] is the characteristic function of the rational numbers, $\chi_\mathbb{Q}$ . It is discontinuous everywhere. However, if you restrict its domain to the irrational numbers, it is continuous. This demonstrates the fact that as n increases, Baire-n functions are more discontinuous.

An important theorem in the theory of Baire function is the Baire Characterisation Theorem. It 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.