Let $f\_w:\mathbb C \to \mathbb C$ be an entire function with $f\_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f\_w$ has infinitely many distinct roots and that $w\mapsto f\_w$ depends real-analytically on $w.$ Does $f\_w$ necessarily have infinitely many distinct roots for all $w \in (0,1)$?



Zeros of entire functions with parameter
$$f\_w(z)=(1-z)e^z+(2w-1)(e^z-1)$$
is a counterexample.


More specifically, the only root of $f\_{1/2}$ is $1$, whereas for each $w\in(0,1)\setminus\{1/2\}$ the function $f\_w$ has infinitely many roots, of the form $2w+W\_k\left(e^{-2 w}(1-2 w)\right)$, where $k$ is any integer and $W\_k$ is the $k$th branch of the [Lambert $W$ function](https://en.wikipedia.org/wiki/Lambert_W_function#Elementary_properties,_branches_and_range).


