Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real solutions to the equation $f(s)=w$ for any real number $w≠0$. Here the set of those real $s$ is a discrete set.
My question is: Can we find a real $u$ such that $f(u)=w$ ($w$ is fixed) and $f'(u)≠0$?
Proof: We can prove this via identity theorem: $f$ cannot have a limit point of zeros in $ℝ$ but $f′$ is also holomorphic, and if (for a contradiction) $f′(s)=0$ whenever $f(s)≠0$ then $f′$ would have a limit point of zeros. Hence a such point $u$ exists.
The answer to the question is no.
Let $f(z)=z\cdot\sin^2(z).$ It is easy to verify that $f$ satisfies the desired conditions. now, $$f'(z)=\sin^2(z)+z\cdot\sin(2z),$$ and clearly, whenever $f(z)=0$, we also have $f'(z)=0.$