Baire function

A Baire function belongs to the smallest hierarchy of real-valued functions containing the continuous functions and closed under pointwise sequential limits.

A real-valued function on a metric space is Baire class one when it is a pointwise limit of continuous functions. On a complete metric space, its continuity points form a dense G-delta set, and its discontinuities form a meagre set. The Baire category theorem also applies after restriction to any nonempty closed subset $F$: the restricted function $f{|}_F$ has a dense G-delta set of relative continuity points in $F$.

Relative continuity here is essential. For the Cantor set $C\subset \mathbb{R}$, its indicator function $1_C$ is Baire class one, since the continuous functionsconverge pointwise to $1_C$. The distance to a set is continuous, which justifies these approximants. Every point of $C$ is a discontinuity of $1_C$ on $\mathbb{R}$, because $C$ has empty interior. Yet $1_C{|}_C$ is the constant continuous function one. Thus relative continuity on a closed subset does not assert ambient continuity at one of its points.

The indicator of the rational numbers in the real numbers is discontinuous everywhere because both the rationals and irrationals are dense. The discontinuity theorem for Baire class one functions therefore shows that it is not Baire class one.