Baire function

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 ${\displaystyle x^2\sin (1/x)}$ 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 ${\displaystyle g_n(x)=max(0,1-nd(x,C))}$, where ${\displaystyle d(x,C)}$ 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 G_δ set (Kechris 1995, Theorem (24.14)).

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, ${\displaystyle {\chi }_{\mathbb{Q}}}$, also known as the Dirichlet function which is discontinuous everywhere.

Proof

We present two proofs.

1. 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 ${\displaystyle g_n(x)=max(0,1-nd(x,K))}$ converges identically to the characteristic function of ${\displaystyle K}$, where ${\displaystyle K}$ is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over ${\displaystyle K_n=\{r_1,r_2,\dots ,r_n\}}$, where ${\displaystyle r_n}$ 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).
2. The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: $${\displaystyle \mathrm{\forall }x\in \mathbb{R},{\chi }_{\mathbb{Q}}(x)=\underset{k\to \mathrm{\infty }}{lim}\left(\underset{j\to \mathrm{\infty }}{lim}{\left(\cos (k!\pi x)\right)}^{2j}\right)}$$ for integer j and k.