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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 in 1899. A Baire set is a set whose characteristic function is a 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.^{[1]}

- 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.

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 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 , where 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 *G*_{δ} set (Kechris 1995, Theorem (24.14)).

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, , also known as the Dirichlet function which is discontinuous everywhere.

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 converges identically to the characteristic function of , where is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over , where 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:

- for integer
*j*and*k*.

**^**Jech, Thomas (November 1981). "The Brave New World of Determinacy".*Bulletin of the American Mathematical Society*.**5**(3): 339–349.

- Baire, René-Louis (1899).
*Sur les fonctions de variables réelles*(Ph.D.) (in French). École Normale Supérieure. - Baire, René-Louis (1905),
*Leçons sur les fonctions discontinues, professées au collège de France*(in French), Gauthier-Villars - Kechris, Alexander S. (1995),
*Classical Descriptive Set Theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, ISBN 978-1-4612-8692-9

- Springer Encyclopaedia of Mathematics article on Baire classes