A function $f:\mathbb R\rightarrow \mathbb R$ is of Baire Class $1$ if there exists a sequence $f_n:\mathbb R\rightarrow \mathbb R$ of continuous functions such that $f_n\rightarrow f$ pointwise.
I know that if $F\subset \mathbb R$ is closed, then $F$ is either countable or $|F|=|\mathbb R|=\mathfrak c$. Therefore, if $f$ is countinuous, then $f^{-1}[\{0\}]$ is either countable or has cardinality $\mathfrak c$. Is this also true if $f$ is of Baire Class 1?
Also, is it true that if $\mathbb R\setminus f^{-1}[\{0\}]$ is uncountable, then it has cardinality $\mathfrak c$? (Supposing that $f$ is of Baire class $1$).
Edit: I noticed that these sets are $G_\delta$'s and $F_\sigma$'s.