 Baire class one function

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.