Let $ f: \mathbb{C} \rightarrow \mathbb{C} $ be an entire function and let $g : \mathbb{C} \rightarrow \mathbb{C} $ be defined by $g(z)= f(z) - f(z+1)$ for all $ z\in \mathbb{C}$. Which of the options are correct :
if $ f(\frac{1}{n}) = 0 $ for all positive integers n, then $f$ is a constant function.
if $ f(n) = 0 $ for all positive integers n, then $f$ is a constant function.
if $ f(\frac{1}{n}) = f(\frac{1}{n}+1)$ for all positive integers n, then $g$ is a constant function.
$ f(n) = f(n+1) $ for all positive integers $n$, then $g$ is a constant function
Please suggest which of the options are correct.
Using the Identity theorem, the options 1 and 3 seem to be correct as in both cases, the sequence of zeros for $\,f\,$ and $\,g\,$ is $ < \frac{1}{n} >$ that converges to zero which belongs to $\Bbb C$. Therefore, in both cases $\,f\,$ and $\,g\,$ are identically equal to zero. But in (2) and (4), we arrive for both $\,f\,$ and $\,g\,$, at the zeros sequence $ <{n}>$ diverges to infinity which does not ensure the required conclusion.