Show that $f$ has infinitely many fixed points

Question

Let $f:ℂ→ℂ$ be an entire function. Assume that the equation $f(s)=a$ has infinitely many real solutions for all reals $a$. Show that $f$ has infinitely many fixed points, i.e., there exists infinitely many reals $t$ such that $f(t)=t$.

Answer 1

It is still false after your update. Take $$f(z) = \frac12 z \sin z.$$ It is clear that $f(x) = a$ has infinitely many real solutions for every real $a$, but the only real solution to $f(x) = x$ is $x=0$.