= 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> $\mathbf1_C$ is Baire class one, since the <continuous functions>
$$
f_n(x)=\max(0,1-n\,d(x,C))
$$
converge <pointwise> to $\mathbf1_C$. The <distance to a set> is <continuous>, which justifies these approximants. Every point of $C$ is a discontinuity of $\mathbf1_C$ on $\mathbb R$, because $C$ has empty <interior>. Yet $\mathbf1_C|_C$ is the constant <continuous function> one. Thus relative <continuity> on a <closed subset> does not assert ambient <continuity> at one of its points.
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