Baire function

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. {{Math proof|title=Proof|drop=hidden|proof=We present two proofs. • 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.}} ==See also==