Baire function

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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[edit]

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 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[edit]

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 x^2\sin(1/x) 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 g_n(x)=\max(0,{1-nd(x,C)}), where d(x,C) 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.

By another theorem of Baire, for every Baire-1 function the points of continuity are a comeager Gδset (Kechris 1995, Theorem (24.14)).

Baire class 2[edit]

Examples:

  • 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.

Baire class 3[edit]

Examples:

See also[edit]

References[edit]

External links[edit]