Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on $X$; it may be assumed to be locally free for simplicity. For any point $y\in Y$ denote by $X_y$ the fiber $f^{-1}(y)$, and by $\mathcal{F}_y$ the pull-back of $\mathcal{F}$ to $X_y$. Fix an integer $i$ and define the function $\phi\colon Y\to \mathbb{Z}$ by $\phi(y):=\dim H^i(X_y,\mathcal{F}_y)$.

**Question.** Is it true that each level set $\phi^{-1}(m)$, $m\in \mathbb{Z}$, can be presented (at least locally) as a finite union of sets of the form $W_1\backslash W_2$, where $W_1,W_2$ are closed analytic sets?

A reference would be very helpful.

**Remark.** An algebraic version of this theorem is true. It follows from the algebraic version of the Grauert semi-continuity theorem which says that the function $\phi$ is upper semi-continuous. While the semi-continuity in Zariski topology does imply the positive answer to my question for algebraic varieties, the semi-continuity in analytic topology apparently does not.