Consider a locally-bounded function $f: W \rightarrow X$, $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$.
Assume that $f$ is Borel measurable
(for every open $O \in \Sigma_X$ ($\sigma$-algebra of $X$) we have $f^{-1}(O) \in \Sigma_W$ ($\sigma$-algebra of $W$))
Consider a locally-bounded discontinuous function $g: X \rightarrow Y$, $Y \subseteq \mathbb{R}^p$.
Say if $\ g \circ f: W \rightarrow Y$ is measurable as well.
If not: counterexample?
If not: under which conditions is $g \circ f$ measurable?