Let $(X,0) \subset \mathbb{C}^n$ be complex analytic set with an isolated singularity. Let $f:(X,0) \to (\mathbb{C},0)$ be a germ of a reduced holomorphic function germ defined on it.
Let $\pi:E \to \mathbb{C}^n$ be an embedded resolution of $X$ with simple normal crossing divisors. Now, precomposing $f$ with $\pi$ outside the critical locus yields a map $f\circ \pi:E \setminus \pi^{-1}(0) \to \mathbb{C}$ which is holomorphic since $\pi$ is biholmorphic outside the divisors.
Can this map be holormophically extended to all $E$? My guess is yes because, (insert your favourite extension theorem here) for example Hartogs extension theorem applies since the divisors have codim $\geq 2$ in $E$.
Now, what are the critical points of $f \circ \pi$ here? all of the divisors? does it have only isolated critical points?