This is a part of a stochastic process exercise.
Here $f$ is not a constant and $\forall s, t, x,$ we have
$$f(s + x)f(t + x) = f(s)f(t).$$
It is required to find all $f$.
If $s = t$, I can get the relation $f(s+x) = \pm f(s)$, so $f$ is a function with two values $\pm z$, where $z$ is a complex constant. Moreover $f$ has some property like periodicity (or translation invariance), but I can't imagine a way to prove it.
Taking $s = 0$ and $t = x$ we get $f(x) f(2x) = f(0) f(x)$, so if $f$ is not $0$ we get $f(2x) = f(0)$. But that says $f$ is constant.