Let $X$ be a compact metric space, and let $f:X\to \mathbb R$ be Baire first class. Can anything be said about $f(X)$?
If $f$ is continuous, then $f(X)$ is compact, but this result does not extend to Baire first-class functions. Indeed for $X=[0,1]$, $$f(x) = \begin{cases} -1 & x=0\\ x & x\neq 0\end{cases}$$ is Baire first class, but $f(X) = \{-1\}\cup (0,1]$ is not compact.
Because a Baire first-class function can be approximated by continuous functions, I would guess that $f(X)$ would be able to be "approximated" by compact sets. Need $f(X)$ be $\sigma$-compact? Does $f(X)$ even need to be Borel measurable?