Baire function
title: "Baire function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["general-topology", "real-analysis", "types-of-functions"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Baire_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 in 1899. A Baire set is a set whose characteristic function is a Baire function.
Classification of Baire functions
Baire functions of class α, for any countable ordinal number α, 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 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 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 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 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 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 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 Gδ set .
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, \chi_\mathbb{Q}, also known as the Dirichlet function which is discontinuous everywhere.
- 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 g_n(x) = \max(0,{1-nd(x,K)}) converges identically to the characteristic function of K, where K is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over K_n = {r_1,r_2,\dots,r_n}, where r_n 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: :::\forall x \in \mathbb R,\quad\chi_{\mathbb Q}(x) = \lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right) :: for integer j and k.}}
References
Inline
General
References
- Jech, Thomas. (November 1981). "The Brave New World of Determinacy". Bulletin of the American Mathematical Society.
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