Baire function

Remarks

The classes increase: $B_\beta \subseteq B_\alpha$ for $\beta \le \alpha$, since a function is the limit of a constant sequence. The set of Baire functions is the smallest set of functions $X \to \R$ that contains the continuous functions and is closed under pointwise limits of sequences. For $X = \R$, the Baire functions are exactly the functions that are measurable for the Borel sigma-algebra.

Example: if $F \colon \R \to \R$ is differentiable, then $F'$ is of Baire class $1$. Indeed $F'(x) = \lim_{n \to \infty} n \big(F(x + 1/n) - F(x)\big)$, and each function on the right is continuous.

Non-example: the Dirichlet function $\mathbf 1_\Q$ is not of Baire class $1$. A function of Baire class $1$ on $\R$ is continuous at the points of a dense set, while $\mathbf 1_\Q$ is continuous nowhere. It is of Baire class $2$, since $\mathbf 1_\Q(x) = \lim_{m \to \infty} \lim_{n \to \infty} \cos(m! \pi x)^{2n}$.