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This particular question was asked in my Quiz yesterday . It is not clear whether the quiz would be discussed or not , so I am asking it here .

Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function such that $f(z+1)=f(z+i)=f(z)$ for every $z$ belonging to $\mathbb{C}$ . Then prove that $f$ is constant .

I will not be able to show an attempt for this as I could not think of anything concise .

Kindly shed some light on how to approach this problem .

Gary
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  • Hint: Liouville's theorem. – Mark Aug 12 '20 at 10:47
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    The hypothesis on $f$ imply that $f$ is bounded. Indeed the image $f( \Bbb C )$ is simply the image of the unit square $$f( { z : 0 \le \Re z , \Im z \le 1 } )$$ which is a compact set, hence bounded. – Crostul Aug 12 '20 at 10:49
  • Construct the set of numbers $z$ of specific type s.t. they have accumulation points in $[0,1]$, then $f(z)-f(0)$ has zeros at these points, and $f(z)-f(0)=0$. – Mike Aug 12 '20 at 12:30

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