Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function:
$$h(s₁,s₂,...,s\_{r+1})=\prod\_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod\_{j=1}^{k}s\_{j}\right)$$
Also, all the $k$-th derivatives of $f$ have infinitely many isolated zeros. Does $h$ have infinitely many **isolated** zeros? I know that $f′(1-2s₁)=0$ has infinitely many isolated zeros. 



Does $h$ have infinitely many isolated zeros?
I assume $f$ was a real-valued analytic function on $(0,1)$, otherwise I do not understand the notation for $h$. But then no zero of $h$ can be isolated. Indeed, the zero set of $h$ is the union of the zero sets of its $r+1$ factors, and all of them vanish on some $r$ dimensional submanifold of $(0,1)^{r+1}$.


