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Liouville Theorem states that: every bounded entire function must be constant.

Now if we know that the function is continuous on the boundary (can it be not continuous on the boundary?) what can we conclude?

(I know that there is something in regard to the max/min but do not fully understand)

gbox
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    I don't understand what you're trying to ask. Which boundary? – Daniel Fischer Jun 13 '17 at 11:42
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    Are you confusing Liouville's Theorem with the Maximum Principle? See also https://math.stackexchange.com/questions/894304/maximum-modulus-principle-implies-liouvilles-theorem. – lhf Jun 13 '17 at 11:44
  • @DanielFischer if we look on an entire and bounded function on $\overline{D}$ – gbox Jun 13 '17 at 11:45
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    @gbox Holomorphic (on an open set $\Omega$) = it has the complex derivative at each point of $\Omega$. Entire = holomorphic on $\Bbb C$. –  Jun 13 '17 at 11:45
  • @lhf yes, that what I was talking about, thanks! – gbox Jun 13 '17 at 11:49
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    $f(z) = e^z$ is holomorphic and bounded on $|z| \le 1$ and it is non-constant. The liouville theorem is if $f(z)$ is holomorphic and bounded on $\mathbb{C}$, then it is constant. – reuns Jun 13 '17 at 14:02

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