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Let $f$ be entire. Suppose $f(z+1)=f(z)=f(z+i)$ for all complex number $z$. Show that $f$ is a constant. Here I tried to use Liouville Theorem.(i.e. If f is entire and bounded, then f must be constant.). But how can I show whether it is bounded?

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Note that for all $z$ we find $w$ with $0\le \Re w\le1$ and $0\le \Im w\le 1$ and $f(z)=f(w)$, and that $f$ is bounded on that square.

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