Lebesgue's characterization of Baire class one functions on $\mathbb R$ is the following:
$f:\mathbb R \rightarrow \mathbb R$ is Baire class one iff for all $r\in \mathbb R$ $\{ x: f > r \}$ and $\{ x: f < r \}$ are $F_\sigma$.
My question is does this same result hold for functions $f:\mathbb R^n \rightarrow \mathbb R$.
My guess is that it does, because I don't see anything in the proof that depends specifically on $n=1$. (see for example p. 15 and on https://www.whitman.edu/Documents/Academics/Mathematics/huh.pdf )