Let $f:\mathbb C\to\mathbb C$ is entire function. And $f(z+1)=f(z)$ and $f(z+i)=f(z)$ then what can you say about $f$?
How to Prove that if $f(z)$ is entire, and $f(z+i) = f(z), f(z+1) = f(z)$, then $f(z)$ is constant?