Suppose $X$ is a complex-analytic variety which is Stein and $f:X\to \mathbb{C}$ is a non-constant analytic function. Denote $X_t=f^{-1}(t)$, and suppose that $X_0$ is singular while $X_t$ for all $t$ such that $|t|\in(0,\epsilon)$ is smooth. Let $\omega$ be a top differential form on $X$ such that $f\omega$ is holomorphic. Does it make sense to compute the residue of $\omega$ along $X_0$?
Background: I'm attempting to apply some methods of Grothendieck Duality for schemes to a problem where I need to deal with some analytic varieties. I am not overly familiar with Grothendieck duality yet and I'm even fuzzier on the connections between the algebraic side and analytic side and what one expects. From what I understand of Grothendieck duality in algebraic geometry, the answer to this question is a yes for local complete intersections(see Hartshorne's Residues and Duality, III.7.3), but I am unfortunately unaware of how similar the analytic machinery is. I suspect that the answer might be yes, given how closely the analytic and algebraic worlds align, but I am very worried about how much I don't know about analytic varieties (my sense is that lots of things that one might take for granted in the algebraic setting go awry in interesting ways).